factorising - numeracy workshop...workshopexpressions and expansion). topics include extracting...
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IntroductionThese slides extend on a basic knowledge of algebra (such as the previous Algebraworkshop Expressions and Expansion). Topics include extracting common factors,factorising quadratic expressions and polynomials.
Workshop resources: These slides are available online:
www.studysmarter.uwa.edu.au → Numeracy and Maths → Online Resources
Next Workshop: See your Workshop Calendar →
www.studysmarter.uwa.edu.au
Drop-in Study Sessions: Monday, Wednesday, Friday, 10am-12pm, Room 2202,Second Floor, Social Sciences South Building, every week.
Email: [email protected]
[email protected] Factorising 3 / 43
IntroductionThese slides extend on a basic knowledge of algebra (such as the previous Algebraworkshop Expressions and Expansion). Topics include extracting common factors,factorising quadratic expressions and polynomials.
Workshop resources: These slides are available online:
www.studysmarter.uwa.edu.au → Numeracy and Maths → Online Resources
Next Workshop: See your Workshop Calendar →
www.studysmarter.uwa.edu.au
Drop-in Study Sessions: Monday, Wednesday, Friday, 10am-12pm, Room 2202,Second Floor, Social Sciences South Building, every week.
Email: [email protected]
[email protected] Factorising 3 / 43
IntroductionThese slides extend on a basic knowledge of algebra (such as the previous Algebraworkshop Expressions and Expansion). Topics include extracting common factors,factorising quadratic expressions and polynomials.
Workshop resources: These slides are available online:
www.studysmarter.uwa.edu.au → Numeracy and Maths → Online Resources
Next Workshop: See your Workshop Calendar →
www.studysmarter.uwa.edu.au
Drop-in Study Sessions: Monday, Wednesday, Friday, 10am-12pm, Room 2202,Second Floor, Social Sciences South Building, every week.
Email: [email protected]
[email protected] Factorising 3 / 43
IntroductionThese slides extend on a basic knowledge of algebra (such as the previous Algebraworkshop Expressions and Expansion). Topics include extracting common factors,factorising quadratic expressions and polynomials.
Workshop resources: These slides are available online:
www.studysmarter.uwa.edu.au → Numeracy and Maths → Online Resources
Next Workshop: See your Workshop Calendar →
www.studysmarter.uwa.edu.au
Drop-in Study Sessions: Monday, Wednesday, Friday, 10am-12pm, Room 2202,Second Floor, Social Sciences South Building, every week.
Email: [email protected]
[email protected] Factorising 3 / 43
IntroductionThese slides extend on a basic knowledge of algebra (such as the previous Algebraworkshop Expressions and Expansion). Topics include extracting common factors,factorising quadratic expressions and polynomials.
Workshop resources: These slides are available online:
www.studysmarter.uwa.edu.au → Numeracy and Maths → Online Resources
Next Workshop: See your Workshop Calendar →
www.studysmarter.uwa.edu.au
Drop-in Study Sessions: Monday, Wednesday, Friday, 10am-12pm, Room 2202,Second Floor, Social Sciences South Building, every week.
Email: [email protected]@uwa.edu.au Factorising 3 / 43
Factors of numbers
A factor of a number is a number that divides into it evenly.
Example: 4 is a factor of 12 since 3× 4 = 12.
Of course, 3 is also a factor of 12. (The others are 1, 2, 6 and 12.)
[email protected] Factorising 4 / 43
Factors of numbers
A factor of a number is a number that divides into it evenly.
Example: 4 is a factor of 12 since 3× 4 = 12.
Of course, 3 is also a factor of 12. (The others are 1, 2, 6 and 12.)
[email protected] Factorising 4 / 43
Factors of numbers
A factor of a number is a number that divides into it evenly.
Example: 4 is a factor of 12 since 3× 4 = 12.
Of course, 3 is also a factor of 12.
(The others are 1, 2, 6 and 12.)
[email protected] Factorising 4 / 43
Factors of numbers
A factor of a number is a number that divides into it evenly.
Example: 4 is a factor of 12 since 3× 4 = 12.
Of course, 3 is also a factor of 12. (The others are
1, 2, 6 and 12.)
[email protected] Factorising 4 / 43
Factors of numbers
A factor of a number is a number that divides into it evenly.
Example: 4 is a factor of 12 since 3× 4 = 12.
Of course, 3 is also a factor of 12. (The others are 1, 2, 6 and 12.)
[email protected] Factorising 4 / 43
Factors of terms
A factor of a term is a number, variable or combination that divides into it evenly.
Example: The term 12x has a factor of 4 since 3x × 4 = 12x .
Important point: the variable x is also a factor of 12x since 12× x = 12x . (Thisis true even though x could be any number, including non-whole numbers.)
Factors of 12x can be made up of combinations of other factors, such as 2, 6x, 1,3x, 12, etc.
[email protected] Factorising 5 / 43
Factors of terms
A factor of a term is a number, variable or combination that divides into it evenly.
Example: The term 12x has a factor of 4 since 3x × 4 = 12x .
Important point: the variable x is also a factor of 12x since 12× x = 12x . (Thisis true even though x could be any number, including non-whole numbers.)
Factors of 12x can be made up of combinations of other factors, such as 2, 6x, 1,3x, 12, etc.
[email protected] Factorising 5 / 43
Factors of terms
A factor of a term is a number, variable or combination that divides into it evenly.
Example: The term 12x has a factor of 4 since 3x × 4 = 12x .
Important point: the variable x is also a factor of 12x since 12× x = 12x . (Thisis true even though x could be any number, including non-whole numbers.)
Factors of 12x can be made up of combinations of other factors, such as 2, 6x, 1,3x, 12, etc.
[email protected] Factorising 5 / 43
Factors of terms
A factor of a term is a number, variable or combination that divides into it evenly.
Example: The term 12x has a factor of 4 since 3x × 4 = 12x .
Important point: the variable x is also a factor of 12x since 12× x = 12x . (Thisis true even though x could be any number, including non-whole numbers.)
Factors of 12x can be made up of combinations of other factors, such as
2, 6x, 1,3x, 12, etc.
[email protected] Factorising 5 / 43
Factors of terms
A factor of a term is a number, variable or combination that divides into it evenly.
Example: The term 12x has a factor of 4 since 3x × 4 = 12x .
Important point: the variable x is also a factor of 12x since 12× x = 12x . (Thisis true even though x could be any number, including non-whole numbers.)
Factors of 12x can be made up of combinations of other factors, such as 2, 6x, 1,3x, 12, etc.
[email protected] Factorising 5 / 43
Factors of expressions
In the previous algebra workshop we looked at the expression
4(2x + 3)
This has two clear factors, 4 and 2x + 3 because
4× (2x + 3) = 4(2x + 3)
When we expanded the brackets, we got
4(2x + 3) = 8x + 12
Our task now is to reverse this process. That is, take an expression like8x + 12 and extract the factors which are common to both terms (8x and 12).
This process is called factorisation.
[email protected] Factorising 6 / 43
Factors of expressions
In the previous algebra workshop we looked at the expression
4(2x + 3)
This has two clear factors,
4 and 2x + 3 because
4× (2x + 3) = 4(2x + 3)
When we expanded the brackets, we got
4(2x + 3) = 8x + 12
Our task now is to reverse this process. That is, take an expression like8x + 12 and extract the factors which are common to both terms (8x and 12).
This process is called factorisation.
[email protected] Factorising 6 / 43
Factors of expressions
In the previous algebra workshop we looked at the expression
4(2x + 3)
This has two clear factors, 4 and
2x + 3 because
4× (2x + 3) = 4(2x + 3)
When we expanded the brackets, we got
4(2x + 3) = 8x + 12
Our task now is to reverse this process. That is, take an expression like8x + 12 and extract the factors which are common to both terms (8x and 12).
This process is called factorisation.
[email protected] Factorising 6 / 43
Factors of expressions
In the previous algebra workshop we looked at the expression
4(2x + 3)
This has two clear factors, 4 and 2x + 3 because
4× (2x + 3) = 4(2x + 3)
When we expanded the brackets, we got
4(2x + 3) = 8x + 12
Our task now is to reverse this process. That is, take an expression like8x + 12 and extract the factors which are common to both terms (8x and 12).
This process is called factorisation.
[email protected] Factorising 6 / 43
Factors of expressions
In the previous algebra workshop we looked at the expression
4(2x + 3)
This has two clear factors, 4 and 2x + 3 because
4× (2x + 3) = 4(2x + 3)
When we expanded the brackets, we got
4(2x + 3) = 8x + 12
Our task now is to reverse this process. That is, take an expression like8x + 12 and extract the factors which are common to both terms (8x and 12).
This process is called factorisation.
[email protected] Factorising 6 / 43
Factors of expressions
In the previous algebra workshop we looked at the expression
4(2x + 3)
This has two clear factors, 4 and 2x + 3 because
4× (2x + 3) = 4(2x + 3)
When we expanded the brackets, we got
4(2x + 3) = 8x + 12
Our task now is to reverse this process.
That is, take an expression like8x + 12 and extract the factors which are common to both terms (8x and 12).
This process is called factorisation.
[email protected] Factorising 6 / 43
Factors of expressions
In the previous algebra workshop we looked at the expression
4(2x + 3)
This has two clear factors, 4 and 2x + 3 because
4× (2x + 3) = 4(2x + 3)
When we expanded the brackets, we got
4(2x + 3) = 8x + 12
Our task now is to reverse this process. That is, take an expression like8x + 12 and extract the factors which are common to both terms (8x and 12).
This process is called factorisation.
[email protected] Factorising 6 / 43
Factors of expressions
In the previous algebra workshop we looked at the expression
4(2x + 3)
This has two clear factors, 4 and 2x + 3 because
4× (2x + 3) = 4(2x + 3)
When we expanded the brackets, we got
4(2x + 3) = 8x + 12
Our task now is to reverse this process. That is, take an expression like8x + 12 and extract the factors which are common to both terms (8x and 12).
This process is called factorisation.
[email protected] Factorising 6 / 43
Factorisation example
Example: Factorise 28x + 98x2.
What factors do the terms 28x and 98x2 have in common?
It might help to imagine some multiplication signs:
28× x and 98× x × x
So, x is a factor common to both terms. Both numbers are even:
2× 14× x and 2× 49× x × x
So, 2 is also a common factor. In fact, this step reveals another one.
2× 2× 7× x and 2× 7× 7× x × x
7 is also a common factor.
[email protected] Factorising 7 / 43
Factorisation example
Example: Factorise 28x + 98x2.
What factors do the terms 28x and 98x2 have in common?
It might help to imagine some multiplication signs:
28× x and 98× x × x
So, x is a factor common to both terms. Both numbers are even:
2× 14× x and 2× 49× x × x
So, 2 is also a common factor. In fact, this step reveals another one.
2× 2× 7× x and 2× 7× 7× x × x
7 is also a common factor.
[email protected] Factorising 7 / 43
Factorisation example
Example: Factorise 28x + 98x2.
What factors do the terms 28x and 98x2 have in common?
It might help to imagine some multiplication signs:
28× x and 98× x × x
So, x is a factor common to both terms. Both numbers are even:
2× 14× x and 2× 49× x × x
So, 2 is also a common factor. In fact, this step reveals another one.
2× 2× 7× x and 2× 7× 7× x × x
7 is also a common factor.
[email protected] Factorising 7 / 43
Factorisation example
Example: Factorise 28x + 98x2.
What factors do the terms 28x and 98x2 have in common?
It might help to imagine some multiplication signs:
28× x and 98× x × x
So, x is a factor common to both terms.
Both numbers are even:
2× 14× x and 2× 49× x × x
So, 2 is also a common factor. In fact, this step reveals another one.
2× 2× 7× x and 2× 7× 7× x × x
7 is also a common factor.
[email protected] Factorising 7 / 43
Factorisation example
Example: Factorise 28x + 98x2.
What factors do the terms 28x and 98x2 have in common?
It might help to imagine some multiplication signs:
28× x and 98× x × x
So, x is a factor common to both terms. Both numbers are even:
2× 14× x and 2× 49× x × x
So, 2 is also a common factor. In fact, this step reveals another one.
2× 2× 7× x and 2× 7× 7× x × x
7 is also a common factor.
[email protected] Factorising 7 / 43
Factorisation example
Example: Factorise 28x + 98x2.
What factors do the terms 28x and 98x2 have in common?
It might help to imagine some multiplication signs:
28× x and 98× x × x
So, x is a factor common to both terms. Both numbers are even:
2× 14× x and 2× 49× x × x
So, 2 is also a common factor. In fact, this step reveals another one.
2× 2× 7× x and 2× 7× 7× x × x
7 is also a common factor.
[email protected] Factorising 7 / 43
Factorisation example
Example: Factorise 28x + 98x2.
What factors do the terms 28x and 98x2 have in common?
It might help to imagine some multiplication signs:
28× x and 98× x × x
So, x is a factor common to both terms. Both numbers are even:
2× 14× x and 2× 49× x × x
So, 2 is also a common factor.
In fact, this step reveals another one.
2× 2× 7× x and 2× 7× 7× x × x
7 is also a common factor.
[email protected] Factorising 7 / 43
Factorisation example
Example: Factorise 28x + 98x2.
What factors do the terms 28x and 98x2 have in common?
It might help to imagine some multiplication signs:
28× x and 98× x × x
So, x is a factor common to both terms. Both numbers are even:
2× 14× x and 2× 49× x × x
So, 2 is also a common factor. In fact, this step reveals another one.
2× 2× 7× x and 2× 7× 7× x × x
7 is also a common factor.
[email protected] Factorising 7 / 43
Factorisation example
Example: Factorise 28x + 98x2.
What factors do the terms 28x and 98x2 have in common?
It might help to imagine some multiplication signs:
28× x and 98× x × x
So, x is a factor common to both terms. Both numbers are even:
2× 14× x and 2× 49× x × x
So, 2 is also a common factor. In fact, this step reveals another one.
2× 2× 7× x and 2× 7× 7× x × x
7 is also a common factor.
[email protected] Factorising 7 / 43
Factorisation example
Example: Factorise 28x + 98x2.
What factors do the terms 28x and 98x2 have in common?
It might help to imagine some multiplication signs:
28× x and 98× x × x
So, x is a factor common to both terms. Both numbers are even:
2× 14× x and 2× 49× x × x
So, 2 is also a common factor. In fact, this step reveals another one.
2× 2× 7× x and 2× 7× 7× x × x
7 is also a common factor.
[email protected] Factorising 7 / 43
Factorisation example
Example: Factorise 28x + 98x2.
2× 2× 7× x and 2× 7× 7× x × x
The combination of common factors is 2× 7× x = 14x .
We call this the highest common factor and write it outside some brackets:
28x + 98x2 = 14x(
2 + 7x
)
The remaining bits of both terms go inside the brackets.
This is the fully factorised form of the original expression.
Note: You can check your answer by expanding the factorised form.
[email protected] Factorising 8 / 43
Factorisation example
Example: Factorise 28x + 98x2.
2× 2× 7× x and 2× 7× 7× x × x
The combination of common factors is 2× 7× x = 14x .
We call this the highest common factor and write it outside some brackets:
28x + 98x2 = 14x(
2 + 7x
)
The remaining bits of both terms go inside the brackets.
This is the fully factorised form of the original expression.
Note: You can check your answer by expanding the factorised form.
[email protected] Factorising 8 / 43
Factorisation example
Example: Factorise 28x + 98x2.
2× 2× 7× x and 2× 7× 7× x × x
The combination of common factors is 2× 7× x = 14x .
We call this the highest common factor and write it outside some brackets:
28x + 98x2 = 14x(
2 + 7x
)
The remaining bits of both terms go inside the brackets.
This is the fully factorised form of the original expression.
Note: You can check your answer by expanding the factorised form.
[email protected] Factorising 8 / 43
Factorisation example
Example: Factorise 28x + 98x2.
2× 2× 7× x and 2× 7× 7× x × x
The combination of common factors is 2× 7× x = 14x .
We call this the highest common factor and write it outside some brackets:
28x + 98x2 = 14x(
2 + 7x
)
The remaining bits of both terms go inside the brackets.
This is the fully factorised form of the original expression.
Note: You can check your answer by expanding the factorised form.
[email protected] Factorising 8 / 43
Factorisation example
Example: Factorise 28x + 98x2.
2× 2× 7× x and 2× 7× 7× x × x
The combination of common factors is 2× 7× x = 14x .
We call this the highest common factor and write it outside some brackets:
28x + 98x2 = 14x(
2 + 7x
)
The remaining bits of both terms go inside the brackets.
This is the fully factorised form of the original expression.
Note: You can check your answer by expanding the factorised form.
[email protected] Factorising 8 / 43
Factorisation example
Example: Factorise 28x + 98x2.
2× 2× 7× x and 2× 7× 7× x × x
The combination of common factors is 2× 7× x = 14x .
We call this the highest common factor and write it outside some brackets:
28x + 98x2 = 14x(2 + 7x)
The remaining bits of both terms go inside the brackets.
This is the fully factorised form of the original expression.
Note: You can check your answer by expanding the factorised form.
[email protected] Factorising 8 / 43
Factorisation example
Example: Factorise 28x + 98x2.
2× 2× 7× x and 2× 7× 7× x × x
The combination of common factors is 2× 7× x = 14x .
We call this the highest common factor and write it outside some brackets:
28x + 98x2 = 14x(2 + 7x)
The remaining bits of both terms go inside the brackets.
This is the fully factorised form of the original expression.
Note: You can check your answer by expanding the factorised form.
[email protected] Factorising 8 / 43
Factorisation example
Example: Factorise 28x + 98x2.
2× 2× 7× x and 2× 7× 7× x × x
The combination of common factors is 2× 7× x = 14x .
We call this the highest common factor and write it outside some brackets:
28x + 98x2 = 14x(2 + 7x)
The remaining bits of both terms go inside the brackets.
This is the fully factorised form of the original expression.
Note: You can check your answer by expanding the factorised form.
[email protected] Factorising 8 / 43
Factorisation exercises
6x − 3x2 =
3x(2− x)
8y + 16y2 = 8y(1 + 2y)
(Note the use of the “hidden” factor of 1 in the first term.)
2x2y + 4x3 = 2x2(y + 2x)
2xy − 4x + 3x2 = x(2y − 4 + 3x)
(Factors must be common to all terms.)
[email protected] Factorising 9 / 43
Factorisation exercises
6x − 3x2 = 3x(
2− x
)
8y + 16y2 = 8y(1 + 2y)
(Note the use of the “hidden” factor of 1 in the first term.)
2x2y + 4x3 = 2x2(y + 2x)
2xy − 4x + 3x2 = x(2y − 4 + 3x)
(Factors must be common to all terms.)
[email protected] Factorising 9 / 43
Factorisation exercises
6x − 3x2 = 3x(2− x)
8y + 16y2 = 8y(1 + 2y)
(Note the use of the “hidden” factor of 1 in the first term.)
2x2y + 4x3 = 2x2(y + 2x)
2xy − 4x + 3x2 = x(2y − 4 + 3x)
(Factors must be common to all terms.)
[email protected] Factorising 9 / 43
Factorisation exercises
6x − 3x2 = 3x(2− x)
8y + 16y2 =
8y(1 + 2y)
(Note the use of the “hidden” factor of 1 in the first term.)
2x2y + 4x3 = 2x2(y + 2x)
2xy − 4x + 3x2 = x(2y − 4 + 3x)
(Factors must be common to all terms.)
[email protected] Factorising 9 / 43
Factorisation exercises
6x − 3x2 = 3x(2− x)
8y + 16y2 = 8y(
1 + 2y
)
(Note the use of the “hidden” factor of 1 in the first term.)
2x2y + 4x3 = 2x2(y + 2x)
2xy − 4x + 3x2 = x(2y − 4 + 3x)
(Factors must be common to all terms.)
[email protected] Factorising 9 / 43
Factorisation exercises
6x − 3x2 = 3x(2− x)
8y + 16y2 = 8y(1 + 2y)
(Note the use of the “hidden” factor of 1 in the first term.)
2x2y + 4x3 = 2x2(y + 2x)
2xy − 4x + 3x2 = x(2y − 4 + 3x)
(Factors must be common to all terms.)
[email protected] Factorising 9 / 43
Factorisation exercises
6x − 3x2 = 3x(2− x)
8y + 16y2 = 8y(1 + 2y)
(Note the use of the “hidden” factor of 1 in the first term.)
2x2y + 4x3 = 2x2(y + 2x)
2xy − 4x + 3x2 = x(2y − 4 + 3x)
(Factors must be common to all terms.)
[email protected] Factorising 9 / 43
Factorisation exercises
6x − 3x2 = 3x(2− x)
8y + 16y2 = 8y(1 + 2y)
(Note the use of the “hidden” factor of 1 in the first term.)
2x2y + 4x3 =
2x2(y + 2x)
2xy − 4x + 3x2 = x(2y − 4 + 3x)
(Factors must be common to all terms.)
[email protected] Factorising 9 / 43
Factorisation exercises
6x − 3x2 = 3x(2− x)
8y + 16y2 = 8y(1 + 2y)
(Note the use of the “hidden” factor of 1 in the first term.)
2x2y + 4x3 = 2x2(
y + 2x
)
2xy − 4x + 3x2 = x(2y − 4 + 3x)
(Factors must be common to all terms.)
[email protected] Factorising 9 / 43
Factorisation exercises
6x − 3x2 = 3x(2− x)
8y + 16y2 = 8y(1 + 2y)
(Note the use of the “hidden” factor of 1 in the first term.)
2x2y + 4x3 = 2x2(y + 2x)
2xy − 4x + 3x2 = x(2y − 4 + 3x)
(Factors must be common to all terms.)
[email protected] Factorising 9 / 43
Factorisation exercises
6x − 3x2 = 3x(2− x)
8y + 16y2 = 8y(1 + 2y)
(Note the use of the “hidden” factor of 1 in the first term.)
2x2y + 4x3 = 2x2(y + 2x)
2xy − 4x + 3x2 =
x(2y − 4 + 3x)
(Factors must be common to all terms.)
[email protected] Factorising 9 / 43
Factorisation exercises
6x − 3x2 = 3x(2− x)
8y + 16y2 = 8y(1 + 2y)
(Note the use of the “hidden” factor of 1 in the first term.)
2x2y + 4x3 = 2x2(y + 2x)
2xy − 4x + 3x2 = x(
2y − 4 + 3x
)
(Factors must be common to all terms.)
[email protected] Factorising 9 / 43
Factorisation exercises
6x − 3x2 = 3x(2− x)
8y + 16y2 = 8y(1 + 2y)
(Note the use of the “hidden” factor of 1 in the first term.)
2x2y + 4x3 = 2x2(y + 2x)
2xy − 4x + 3x2 = x(2y − 4 + 3x)
(Factors must be common to all terms.)
[email protected] Factorising 9 / 43
Factorisation exercises
6x − 3x2 = 3x(2− x)
8y + 16y2 = 8y(1 + 2y)
(Note the use of the “hidden” factor of 1 in the first term.)
2x2y + 4x3 = 2x2(y + 2x)
2xy − 4x + 3x2 = x(2y − 4 + 3x)
(Factors must be common to all terms.)
[email protected] Factorising 9 / 43
Double bracket expressions
Here is a common type of double bracket expression.
Let’s expand the brackets tosee whether there are any useful patterns we can use to reverse the process:
(x + 2)(x + 5) =
x2 + 5x+ 2x + 2× 5
= x2 + (5 + 2)x + 2× 5
= x2 + 7x + 10
The pattern is here: 5 + 2 = 7 and 2× 5 = 10.
[email protected] Factorising 10 / 43
Double bracket expressions
Here is a common type of double bracket expression. Let’s expand the brackets tosee whether there are any useful patterns we can use to reverse the process:
(x + 2)(x + 5) =
x2 + 5x+ 2x + 2× 5
= x2 + (5 + 2)x + 2× 5
= x2 + 7x + 10
The pattern is here: 5 + 2 = 7 and 2× 5 = 10.
[email protected] Factorising 10 / 43
Double bracket expressions
Here is a common type of double bracket expression. Let’s expand the brackets tosee whether there are any useful patterns we can use to reverse the process:
(x + 2)(x + 5) =
x2 + 5x+ 2x + 2× 5
= x2 + (5 + 2)x + 2× 5
= x2 + 7x + 10
The pattern is here: 5 + 2 = 7 and 2× 5 = 10.
[email protected] Factorising 10 / 43
Double bracket expressions
Here is a common type of double bracket expression. Let’s expand the brackets tosee whether there are any useful patterns we can use to reverse the process:
(x + 2)(x + 5) = x2
+ 5x+ 2x + 2× 5
= x2 + (5 + 2)x + 2× 5
= x2 + 7x + 10
The pattern is here: 5 + 2 = 7 and 2× 5 = 10.
[email protected] Factorising 10 / 43
Double bracket expressions
Here is a common type of double bracket expression. Let’s expand the brackets tosee whether there are any useful patterns we can use to reverse the process:
(x + 2)(x + 5) = x2
+ 5x+ 2x + 2× 5
= x2 + (5 + 2)x + 2× 5
= x2 + 7x + 10
The pattern is here: 5 + 2 = 7 and 2× 5 = 10.
[email protected] Factorising 10 / 43
Double bracket expressions
Here is a common type of double bracket expression. Let’s expand the brackets tosee whether there are any useful patterns we can use to reverse the process:
(x + 2)(x + 5) = x2 + 5x
+ 2x + 2× 5
= x2 + (5 + 2)x + 2× 5
= x2 + 7x + 10
The pattern is here: 5 + 2 = 7 and 2× 5 = 10.
[email protected] Factorising 10 / 43
Double bracket expressions
Here is a common type of double bracket expression. Let’s expand the brackets tosee whether there are any useful patterns we can use to reverse the process:
(x + 2)(x + 5) = x2 + 5x
+ 2x + 2× 5
= x2 + (5 + 2)x + 2× 5
= x2 + 7x + 10
The pattern is here: 5 + 2 = 7 and 2× 5 = 10.
[email protected] Factorising 10 / 43
Double bracket expressions
Here is a common type of double bracket expression. Let’s expand the brackets tosee whether there are any useful patterns we can use to reverse the process:
(x + 2)(x + 5) = x2 + 5x+ 2x
+ 2× 5
= x2 + (5 + 2)x + 2× 5
= x2 + 7x + 10
The pattern is here: 5 + 2 = 7 and 2× 5 = 10.
[email protected] Factorising 10 / 43
Double bracket expressions
Here is a common type of double bracket expression. Let’s expand the brackets tosee whether there are any useful patterns we can use to reverse the process:
(x + 2)(x + 5) = x2 + 5x+ 2x
+ 2× 5
= x2 + (5 + 2)x + 2× 5
= x2 + 7x + 10
The pattern is here: 5 + 2 = 7 and 2× 5 = 10.
[email protected] Factorising 10 / 43
Double bracket expressions
Here is a common type of double bracket expression. Let’s expand the brackets tosee whether there are any useful patterns we can use to reverse the process:
(x + 2)(x + 5) = x2 + 5x+ 2x + 2× 5
= x2 + (5 + 2)x + 2× 5
= x2 + 7x + 10
The pattern is here: 5 + 2 = 7 and 2× 5 = 10.
[email protected] Factorising 10 / 43
Double bracket expressions
Here is a common type of double bracket expression. Let’s expand the brackets tosee whether there are any useful patterns we can use to reverse the process:
(x + 2)(x + 5) = x2 + 5x+ 2x + 2× 5
= x2 + (5 + 2)x + 2× 5
= x2 + 7x + 10
The pattern is here: 5 + 2 = 7 and 2× 5 = 10.
[email protected] Factorising 10 / 43
Double bracket expressions
Here is a common type of double bracket expression. Let’s expand the brackets tosee whether there are any useful patterns we can use to reverse the process:
(x + 2)(x + 5) = x2 + 5x+ 2x + 2× 5
= x2 + (5 + 2)x + 2× 5
= x2 + 7x + 10
The pattern is here: 5 + 2 = 7 and 2× 5 = 10.
[email protected] Factorising 10 / 43
Double bracket expressions
Here is a common type of double bracket expression. Let’s expand the brackets tosee whether there are any useful patterns we can use to reverse the process:
(x + 2)(x + 5) = x2 + 5x+ 2x + 2× 5
= x2 + (5 + 2)x + 2× 5
= x2 + 7x + 10
The pattern is here: 5 + 2 = 7 and 2× 5 = 10.
[email protected] Factorising 10 / 43
A Rule
So, in general, whenever we are asked to expand an expression of the form:
(x + a)(x + b)
we always end up with
x2 + (a + b)x + ab
That is, you always add the two numbers a and b together to get the numbermultiplying x , and you multiply them to get the constant term.
[email protected] Factorising 11 / 43
A Rule
So, in general, whenever we are asked to expand an expression of the form:
(x + a)(x + b)
we always end up with
x2 + (a + b)x + ab
That is, you always add the two numbers a and b together to get the numbermultiplying x , and you multiply them to get the constant term.
[email protected] Factorising 11 / 43
Factorisation
If we are asked to factorise:
x2 + 5x + 6
then we are being asked to write the above in the form
(x + a)(x + b)
where a + b = 5 and a× b = 6. Can you find two numbers which do this?
The answer is 2 and 3 (2 + 3 = 5, 2× 3 = 6).
So we can factorise the expression as follows:
x2 + 5x + 6 = (x + 2)(x + 3)
[email protected] Factorising 12 / 43
Factorisation
If we are asked to factorise:
x2 + 5x + 6
then we are being asked to write the above in the form
(x + a)(x + b)
where a + b = 5 and a× b = 6. Can you find two numbers which do this?
The answer is 2 and 3 (2 + 3 = 5, 2× 3 = 6).
So we can factorise the expression as follows:
x2 + 5x + 6 = (x + 2)(x + 3)
[email protected] Factorising 12 / 43
Factorisation
If we are asked to factorise:
x2 + 5x + 6
then we are being asked to write the above in the form
(x + a)(x + b)
where a + b = 5 and a× b = 6. Can you find two numbers which do this?
The answer is 2 and 3 (2 + 3 = 5, 2× 3 = 6).
So we can factorise the expression as follows:
x2 + 5x + 6 = (x + 2)(x + 3)
[email protected] Factorising 12 / 43
Factorisation
If we are asked to factorise:
x2 + 5x + 6
then we are being asked to write the above in the form
(x + a)(x + b)
where a + b = 5 and a× b = 6. Can you find two numbers which do this?
The answer is 2 and 3 (2 + 3 = 5, 2× 3 = 6).
So we can factorise the expression as follows:
x2 + 5x + 6 = (x + 2)(x + 3)
[email protected] Factorising 12 / 43
Factorisation
Factorise x2 − 3x − 10
So x2 − 3x − 10 = (x + a)(x + b)
where a + b = −3 and a× b = −10. Which two numbers do this?
The answer is −5 and 2 (−5 + 2 = −3, −5× 2 = −10).
So we can factorise the expression as follows:
x2 − 3x − 10 = (x − 5)(x + 2)
[email protected] Factorising 13 / 43
Factorisation
Factorise x2 − 3x − 10
So x2 − 3x − 10 = (x + a)(x + b)
where a + b = −3 and a× b = −10. Which two numbers do this?
The answer is −5 and 2 (−5 + 2 = −3, −5× 2 = −10).
So we can factorise the expression as follows:
x2 − 3x − 10 = (x − 5)(x + 2)
[email protected] Factorising 13 / 43
Factorisation
Factorise x2 − 3x − 10
So x2 − 3x − 10 = (x + a)(x + b)
where a + b = −3 and a× b = −10. Which two numbers do this?
The answer is −5 and 2 (−5 + 2 = −3, −5× 2 = −10).
So we can factorise the expression as follows:
x2 − 3x − 10 = (x − 5)(x + 2)
[email protected] Factorising 13 / 43
Factorisation
Factorise x2 − 3x − 10
So x2 − 3x − 10 = (x + a)(x + b)
where a + b = −3 and a× b = −10. Which two numbers do this?
The answer is −5 and 2 (−5 + 2 = −3, −5× 2 = −10).
So we can factorise the expression as follows:
x2 − 3x − 10 = (x − 5)(x + 2)
[email protected] Factorising 13 / 43
Removing Factors
We have seen a method which usually works when the multiplier of x2 isequal to 1.
What if we were asked to factorise:
3x2 + 9x + 6
We notice that the multipliers of all three terms in the above expression aredivisible by 3. Hence, we can factor out this common factor as follows:
3x2 + 9x + 6 = 3(x2 + 3x + 2)
Now, the expression in brackets is just like we saw on the previous slides, we lookfor two numbers which add to 3 and multiply to 2. This gives us:
3x2 + 9x + 6 = 3(x + 1)(x + 2)
[email protected] Factorising 14 / 43
Removing Factors
We have seen a method which usually works when the multiplier of x2 isequal to 1.
What if we were asked to factorise:
3x2 + 9x + 6
We notice that the multipliers of all three terms in the above expression aredivisible by 3. Hence, we can factor out this common factor as follows:
3x2 + 9x + 6 = 3(x2 + 3x + 2)
Now, the expression in brackets is just like we saw on the previous slides, we lookfor two numbers which add to 3 and multiply to 2. This gives us:
3x2 + 9x + 6 = 3(x + 1)(x + 2)
[email protected] Factorising 14 / 43
Removing Factors
We have seen a method which usually works when the multiplier of x2 isequal to 1.
What if we were asked to factorise:
3x2 + 9x + 6
We notice that the multipliers of all three terms in the above expression aredivisible by 3. Hence, we can factor out this common factor as follows:
3x2 + 9x + 6 = 3(x2 + 3x + 2)
Now, the expression in brackets is just like we saw on the previous slides, we lookfor two numbers which add to 3 and multiply to 2. This gives us:
3x2 + 9x + 6 = 3
(x + 1)(x + 2)
[email protected] Factorising 14 / 43
Removing Factors
We have seen a method which usually works when the multiplier of x2 isequal to 1.
What if we were asked to factorise:
3x2 + 9x + 6
We notice that the multipliers of all three terms in the above expression aredivisible by 3. Hence, we can factor out this common factor as follows:
3x2 + 9x + 6 = 3(x2 + 3x + 2)
Now, the expression in brackets is just like we saw on the previous slides, we lookfor two numbers which add to 3 and multiply to 2. This gives us:
3x2 + 9x + 6 = 3(x + 1)(x + 2)
[email protected] Factorising 14 / 43
Removing Factors
Factorise the following expression.
5x2 + 40x + 60
Here we see that the multiplier of x2 is 5. We also notice that all multipliers inthe above expression are divisible by 5. Hence, we can factor out this common
factor as follows:
5x2 + 40x + 60 = 5(x2 + 8x + 12)
Now we look for two numbers which add to 8 and multiply to 12. This gives us:
5x2 + 40x + 60 = 5(x + 2)(x + 6)
[email protected] Factorising 15 / 43
Removing Factors
Factorise the following expression.
5x2 + 40x + 60
Here we see that the multiplier of x2 is 5. We also notice that all multipliers inthe above expression are divisible by 5. Hence, we can factor out this common
factor as follows:
5x2 + 40x + 60 = 5(x2 + 8x + 12)
Now we look for two numbers which add to 8 and multiply to 12. This gives us:
5x2 + 40x + 60 = 5(x + 2)(x + 6)
[email protected] Factorising 15 / 43
Removing Factors
Factorise the following expression.
5x2 + 40x + 60
Here we see that the multiplier of x2 is 5. We also notice that all multipliers inthe above expression are divisible by 5. Hence, we can factor out this common
factor as follows:
5x2 + 40x + 60 = 5(x2 + 8x + 12)
Now we look for two numbers which add to 8 and multiply to 12. This gives us:
5x2 + 40x + 60 = 5
(x + 2)(x + 6)
[email protected] Factorising 15 / 43
Removing Factors
Factorise the following expression.
5x2 + 40x + 60
Here we see that the multiplier of x2 is 5. We also notice that all multipliers inthe above expression are divisible by 5. Hence, we can factor out this common
factor as follows:
5x2 + 40x + 60 = 5(x2 + 8x + 12)
Now we look for two numbers which add to 8 and multiply to 12. This gives us:
5x2 + 40x + 60 = 5(x + 2)(x + 6)
[email protected] Factorising 15 / 43
Factorisation
If we are asked to factorise:
x2 − 9
then we are being asked to write the above in the form
(x + a)(x + b)
It looks different to the expressions we have been factorising because it appears tohave no x term. However, we can make it the same if we include an x term with a
multiplier of 0:
x2+0x − 9
So we know that we need two numbers which add up to 0 and multiply to give−9. Can you find two numbers which do this?
The answer is 3 and −3 (3 +−3 = 0, 3×−3 = −9).
So we can factorise the expression as follows:
x2 − 9 = (x + 3)(x − 3)
[email protected] Factorising 16 / 43
Factorisation
If we are asked to factorise:
x2 − 9
then we are being asked to write the above in the form
(x + a)(x + b)
It looks different to the expressions we have been factorising because it appears tohave no x term.
However, we can make it the same if we include an x term with amultiplier of 0:
x2+0x − 9
So we know that we need two numbers which add up to 0 and multiply to give−9. Can you find two numbers which do this?
The answer is 3 and −3 (3 +−3 = 0, 3×−3 = −9).
So we can factorise the expression as follows:
x2 − 9 = (x + 3)(x − 3)
[email protected] Factorising 16 / 43
Factorisation
If we are asked to factorise:
x2 − 9
then we are being asked to write the above in the form
(x + a)(x + b)
It looks different to the expressions we have been factorising because it appears tohave no x term. However, we can make it the same if we include an x term with a
multiplier of
0:
x2+0x − 9
So we know that we need two numbers which add up to 0 and multiply to give−9. Can you find two numbers which do this?
The answer is 3 and −3 (3 +−3 = 0, 3×−3 = −9).
So we can factorise the expression as follows:
x2 − 9 = (x + 3)(x − 3)
[email protected] Factorising 16 / 43
Factorisation
If we are asked to factorise:
x2 − 9
then we are being asked to write the above in the form
(x + a)(x + b)
It looks different to the expressions we have been factorising because it appears tohave no x term. However, we can make it the same if we include an x term with a
multiplier of 0:
x2+0x − 9
So we know that we need two numbers which add up to 0 and multiply to give−9. Can you find two numbers which do this?
The answer is 3 and −3 (3 +−3 = 0, 3×−3 = −9).
So we can factorise the expression as follows:
x2 − 9 = (x + 3)(x − 3)
[email protected] Factorising 16 / 43
Factorisation
If we are asked to factorise:
x2 − 9
then we are being asked to write the above in the form
(x + a)(x + b)
It looks different to the expressions we have been factorising because it appears tohave no x term. However, we can make it the same if we include an x term with a
multiplier of 0:
x2+0x − 9
So we know that we need two numbers which add up to 0 and multiply to give−9. Can you find two numbers which do this?
The answer is 3 and −3 (3 +−3 = 0, 3×−3 = −9).
So we can factorise the expression as follows:
x2 − 9 = (x + 3)(x − 3)
[email protected] Factorising 16 / 43
Factorisation
If we are asked to factorise:
x2 − 9
then we are being asked to write the above in the form
(x + a)(x + b)
It looks different to the expressions we have been factorising because it appears tohave no x term. However, we can make it the same if we include an x term with a
multiplier of 0:
x2+0x − 9
So we know that we need two numbers which add up to 0 and multiply to give−9. Can you find two numbers which do this?
The answer is 3 and −3 (3 +−3 = 0, 3×−3 = −9).
So we can factorise the expression as follows:
x2 − 9 = (x + 3)(x − 3)
[email protected] Factorising 16 / 43
Factorisation
If we are asked to factorise:
x2 − 9
then we are being asked to write the above in the form
(x + a)(x + b)
It looks different to the expressions we have been factorising because it appears tohave no x term. However, we can make it the same if we include an x term with a
multiplier of 0:
x2+0x − 9
So we know that we need two numbers which add up to 0 and multiply to give−9. Can you find two numbers which do this?
The answer is 3 and −3 (3 +−3 = 0, 3×−3 = −9).
So we can factorise the expression as follows:
x2 − 9 = (x + 3)(x − 3)
[email protected] Factorising 16 / 43
The Difference of Two Squares
If you are asked to factorise x2 − 16 we get:
(x + 4)(x − 4)
If you are asked to factorise x2 − 36 we get:
(x + 6)(x − 6)
This leads us to a general formula called the difference of two squares:
x2 − a2 = (x + a)(x − a)
Tip: Try to avoid memorising too many formulas. It’s handy to remember thatthis is just a simple special case of a more general process.
[email protected] Factorising 17 / 43
The Difference of Two Squares
If you are asked to factorise x2 − 16 we get:
(x + 4)(x − 4)
If you are asked to factorise x2 − 36 we get:
(x + 6)(x − 6)
This leads us to a general formula called the difference of two squares:
x2 − a2 = (x + a)(x − a)
Tip: Try to avoid memorising too many formulas. It’s handy to remember thatthis is just a simple special case of a more general process.
[email protected] Factorising 17 / 43
The Difference of Two Squares
If you are asked to factorise x2 − 16 we get:
(x + 4)(x − 4)
If you are asked to factorise x2 − 36 we get:
(x + 6)(x − 6)
This leads us to a general formula called the difference of two squares:
x2 − a2 = (x + a)(x − a)
Tip: Try to avoid memorising too many formulas. It’s handy to remember thatthis is just a simple special case of a more general process.
[email protected] Factorising 17 / 43
The Difference of Two Squares
If you are asked to factorise x2 − 16 we get:
(x + 4)(x − 4)
If you are asked to factorise x2 − 36 we get:
(x + 6)(x − 6)
This leads us to a general formula called the difference of two squares:
x2 − a2 = (x + a)(x − a)
Tip: Try to avoid memorising too many formulas. It’s handy to remember thatthis is just a simple special case of a more general process.
[email protected] Factorising 17 / 43
The Difference of Two Squares
If you are asked to factorise x2 − 16 we get:
(x + 4)(x − 4)
If you are asked to factorise x2 − 36 we get:
(x + 6)(x − 6)
This leads us to a general formula called the difference of two squares:
x2 − a2 = (x + a)(x − a)
Tip: Try to avoid memorising too many formulas. It’s handy to remember thatthis is just a simple special case of a more general process.
[email protected] Factorising 17 / 43
The Difference of Two Squares
If you are asked to factorise x2 − 16 we get:
(x + 4)(x − 4)
If you are asked to factorise x2 − 36 we get:
(x + 6)(x − 6)
This leads us to a general formula called the difference of two squares:
x2 − a2 = (x + a)(x − a)
Tip: Try to avoid memorising too many formulas. It’s handy to remember thatthis is just a simple special case of a more general process.
[email protected] Factorising 17 / 43
The Difference of Two Squares: Examples
x2 − 25 =
(x − 5)(x + 5)
x2 − 49 = (x + 7)(x − 7)
4x2 − 25 = (2x)2 − 52
= (2x − 5)(2x + 5)
9x4 − 64 =(3x2
)2 − 82
=(3x2 − 8
) (3x2 + 8
)
[email protected] Factorising 18 / 43
The Difference of Two Squares: Examples
x2 − 25 = (x − 5)(x + 5)
x2 − 49 = (x + 7)(x − 7)
4x2 − 25 = (2x)2 − 52
= (2x − 5)(2x + 5)
9x4 − 64 =(3x2
)2 − 82
=(3x2 − 8
) (3x2 + 8
)
[email protected] Factorising 18 / 43
The Difference of Two Squares: Examples
x2 − 25 = (x − 5)(x + 5)
x2 − 49 =
(x + 7)(x − 7)
4x2 − 25 = (2x)2 − 52
= (2x − 5)(2x + 5)
9x4 − 64 =(3x2
)2 − 82
=(3x2 − 8
) (3x2 + 8
)
[email protected] Factorising 18 / 43
The Difference of Two Squares: Examples
x2 − 25 = (x − 5)(x + 5)
x2 − 49 = (x + 7)(x − 7)
4x2 − 25 = (2x)2 − 52
= (2x − 5)(2x + 5)
9x4 − 64 =(3x2
)2 − 82
=(3x2 − 8
) (3x2 + 8
)
[email protected] Factorising 18 / 43
The Difference of Two Squares: Examples
x2 − 25 = (x − 5)(x + 5)
x2 − 49 = (x + 7)(x − 7)
4x2 − 25 =
(2x)2 − 52
= (2x − 5)(2x + 5)
9x4 − 64 =(3x2
)2 − 82
=(3x2 − 8
) (3x2 + 8
)
[email protected] Factorising 18 / 43
The Difference of Two Squares: Examples
x2 − 25 = (x − 5)(x + 5)
x2 − 49 = (x + 7)(x − 7)
4x2 − 25 = (2x)2 − 52
=
(2x − 5)(2x + 5)
9x4 − 64 =(3x2
)2 − 82
=(3x2 − 8
) (3x2 + 8
)
[email protected] Factorising 18 / 43
The Difference of Two Squares: Examples
x2 − 25 = (x − 5)(x + 5)
x2 − 49 = (x + 7)(x − 7)
4x2 − 25 = (2x)2 − 52
= (2x − 5)(2x + 5)
9x4 − 64 =(3x2
)2 − 82
=(3x2 − 8
) (3x2 + 8
)
[email protected] Factorising 18 / 43
The Difference of Two Squares: Examples
x2 − 25 = (x − 5)(x + 5)
x2 − 49 = (x + 7)(x − 7)
4x2 − 25 = (2x)2 − 52
= (2x − 5)(2x + 5)
9x4 − 64 =
(3x2
)2 − 82
=(3x2 − 8
) (3x2 + 8
)
[email protected] Factorising 18 / 43
The Difference of Two Squares: Examples
x2 − 25 = (x − 5)(x + 5)
x2 − 49 = (x + 7)(x − 7)
4x2 − 25 = (2x)2 − 52
= (2x − 5)(2x + 5)
9x4 − 64 =(3x2
)2 − 82
=
(3x2 − 8
) (3x2 + 8
)
[email protected] Factorising 18 / 43
The Difference of Two Squares: Examples
x2 − 25 = (x − 5)(x + 5)
x2 − 49 = (x + 7)(x − 7)
4x2 − 25 = (2x)2 − 52
= (2x − 5)(2x + 5)
9x4 − 64 =(3x2
)2 − 82
=(3x2 − 8
) (3x2 + 8
)
[email protected] Factorising 18 / 43
The Difference of Two Squares: Examples
Sometimes we need to factor out the highest common factor:
3x2 − 75 =
3(x2 − 25)
= 3(x + 5)(x − 5)
2x2 − 8 = 2(x2 − 4)
= 2(x + 2)(x − 2)
x3 − 25x = x(x2 − 25)
= x(x − 5)(x + 5)
18x3 − 32x = 2x(9x2 − 16)
= 2x(3x + 4)(3x − 4)
[email protected] Factorising 19 / 43
The Difference of Two Squares: Examples
Sometimes we need to factor out the highest common factor:
3x2 − 75 = 3(x2 − 25)
= 3
(x + 5)(x − 5)
2x2 − 8 = 2(x2 − 4)
= 2(x + 2)(x − 2)
x3 − 25x = x(x2 − 25)
= x(x − 5)(x + 5)
18x3 − 32x = 2x(9x2 − 16)
= 2x(3x + 4)(3x − 4)
[email protected] Factorising 19 / 43
The Difference of Two Squares: Examples
Sometimes we need to factor out the highest common factor:
3x2 − 75 = 3(x2 − 25)
= 3(x + 5)(x − 5)
2x2 − 8 = 2(x2 − 4)
= 2(x + 2)(x − 2)
x3 − 25x = x(x2 − 25)
= x(x − 5)(x + 5)
18x3 − 32x = 2x(9x2 − 16)
= 2x(3x + 4)(3x − 4)
[email protected] Factorising 19 / 43
The Difference of Two Squares: Examples
Sometimes we need to factor out the highest common factor:
3x2 − 75 = 3(x2 − 25)
= 3(x + 5)(x − 5)
2x2 − 8 =
2(x2 − 4)
= 2(x + 2)(x − 2)
x3 − 25x = x(x2 − 25)
= x(x − 5)(x + 5)
18x3 − 32x = 2x(9x2 − 16)
= 2x(3x + 4)(3x − 4)
[email protected] Factorising 19 / 43
The Difference of Two Squares: Examples
Sometimes we need to factor out the highest common factor:
3x2 − 75 = 3(x2 − 25)
= 3(x + 5)(x − 5)
2x2 − 8 = 2(x2 − 4)
= 2
(x + 2)(x − 2)
x3 − 25x = x(x2 − 25)
= x(x − 5)(x + 5)
18x3 − 32x = 2x(9x2 − 16)
= 2x(3x + 4)(3x − 4)
[email protected] Factorising 19 / 43
The Difference of Two Squares: Examples
Sometimes we need to factor out the highest common factor:
3x2 − 75 = 3(x2 − 25)
= 3(x + 5)(x − 5)
2x2 − 8 = 2(x2 − 4)
= 2(x + 2)(x − 2)
x3 − 25x = x(x2 − 25)
= x(x − 5)(x + 5)
18x3 − 32x = 2x(9x2 − 16)
= 2x(3x + 4)(3x − 4)
[email protected] Factorising 19 / 43
The Difference of Two Squares: Examples
Sometimes we need to factor out the highest common factor:
3x2 − 75 = 3(x2 − 25)
= 3(x + 5)(x − 5)
2x2 − 8 = 2(x2 − 4)
= 2(x + 2)(x − 2)
x3 − 25x =
x(x2 − 25)
= x(x − 5)(x + 5)
18x3 − 32x = 2x(9x2 − 16)
= 2x(3x + 4)(3x − 4)
[email protected] Factorising 19 / 43
The Difference of Two Squares: Examples
Sometimes we need to factor out the highest common factor:
3x2 − 75 = 3(x2 − 25)
= 3(x + 5)(x − 5)
2x2 − 8 = 2(x2 − 4)
= 2(x + 2)(x − 2)
x3 − 25x = x(x2 − 25)
= x
(x − 5)(x + 5)
18x3 − 32x = 2x(9x2 − 16)
= 2x(3x + 4)(3x − 4)
[email protected] Factorising 19 / 43
The Difference of Two Squares: Examples
Sometimes we need to factor out the highest common factor:
3x2 − 75 = 3(x2 − 25)
= 3(x + 5)(x − 5)
2x2 − 8 = 2(x2 − 4)
= 2(x + 2)(x − 2)
x3 − 25x = x(x2 − 25)
= x(x − 5)(x + 5)
18x3 − 32x = 2x(9x2 − 16)
= 2x(3x + 4)(3x − 4)
[email protected] Factorising 19 / 43
The Difference of Two Squares: Examples
Sometimes we need to factor out the highest common factor:
3x2 − 75 = 3(x2 − 25)
= 3(x + 5)(x − 5)
2x2 − 8 = 2(x2 − 4)
= 2(x + 2)(x − 2)
x3 − 25x = x(x2 − 25)
= x(x − 5)(x + 5)
18x3 − 32x =
2x(9x2 − 16)
= 2x(3x + 4)(3x − 4)
[email protected] Factorising 19 / 43
The Difference of Two Squares: Examples
Sometimes we need to factor out the highest common factor:
3x2 − 75 = 3(x2 − 25)
= 3(x + 5)(x − 5)
2x2 − 8 = 2(x2 − 4)
= 2(x + 2)(x − 2)
x3 − 25x = x(x2 − 25)
= x(x − 5)(x + 5)
18x3 − 32x = 2x(9x2 − 16)
= 2x
(3x + 4)(3x − 4)
[email protected] Factorising 19 / 43
The Difference of Two Squares: Examples
Sometimes we need to factor out the highest common factor:
3x2 − 75 = 3(x2 − 25)
= 3(x + 5)(x − 5)
2x2 − 8 = 2(x2 − 4)
= 2(x + 2)(x − 2)
x3 − 25x = x(x2 − 25)
= x(x − 5)(x + 5)
18x3 − 32x = 2x(9x2 − 16)
= 2x(3x + 4)(3x − 4)
[email protected] Factorising 19 / 43
Harder Factorisation
What if we can’t easily factor out the multiplier of x2?
First, consider expanding
(7x + 4)(3x + 5)
If we do this we get
21x2 + 35x + 12x + 20
which then equals
21x2 + 47x + 20
Expansion is easy, but doing this problem backwards is tricky if we don’t knowwhere we started. Our previous methods don’t work here.
[email protected] Factorising 20 / 43
Harder Factorisation
What if we can’t easily factor out the multiplier of x2?
First, consider expanding
(7x + 4)(3x + 5)
If we do this we get
21x2 + 35x + 12x + 20
which then equals
21x2 + 47x + 20
Expansion is easy, but doing this problem backwards is tricky if we don’t knowwhere we started. Our previous methods don’t work here.
[email protected] Factorising 20 / 43
Harder Factorisation
What if we can’t easily factor out the multiplier of x2?
First, consider expanding
(7x + 4)(3x + 5)
If we do this we get
21x2 + 35x + 12x + 20
which then equals
21x2 + 47x + 20
Expansion is easy, but doing this problem backwards is tricky if we don’t knowwhere we started. Our previous methods don’t work here.
[email protected] Factorising 20 / 43
Harder Factorisation
What if we can’t easily factor out the multiplier of x2?
First, consider expanding
(7x + 4)(3x + 5)
If we do this we get
21x2 + 35x + 12x + 20
which then equals
21x2 + 47x + 20
Expansion is easy, but doing this problem backwards is tricky if we don’t knowwhere we started. Our previous methods don’t work here.
[email protected] Factorising 20 / 43
Harder Factorisation
What if we can’t easily factor out the multiplier of x2?
First, consider expanding
(7x + 4)(3x + 5)
If we do this we get
21x2 + 35x + 12x + 20
which then equals
21x2 + 47x + 20
Expansion is easy, but doing this problem backwards is tricky if we don’t knowwhere we started. Our previous methods don’t work here.
[email protected] Factorising 20 / 43
Harder Factorisation
In general, when we see an expression of the form
Ax2 + Bx + C
we want to factorise it by writing it in the following form:
(ax + b)(cx + d)
Note that a and c multiply to ptoduce the multiplier of x2 (A), and that b and dmultiply to produce the constant on the end (C).
Then we need to play around with it a bit.
[email protected] Factorising 21 / 43
Harder Factorisation
In general, when we see an expression of the form
Ax2 + Bx + C
we want to factorise it by writing it in the following form:
(ax + b)(cx + d)
Note that a and c multiply to ptoduce the multiplier of x2 (A), and that b and dmultiply to produce the constant on the end (C).
Then we need to play around with it a bit.
[email protected] Factorising 21 / 43
Harder Factorisation
In general, when we see an expression of the form
Ax2 + Bx + C
we want to factorise it by writing it in the following form:
(ax + b)(cx + d)
Note that a and c multiply to ptoduce the multiplier of x2 (A), and that b and dmultiply to produce the constant on the end (C).
Then we need to play around with it a bit.
[email protected] Factorising 21 / 43
Harder Factorisation: Example
Factorise 2x2 + 3x + 1.
We need to write this in the form (ax + b)(cx + d).
The numbers a and c must multiply up to 2, and so one of them must be 1 andthe other must be 2. (It doesn’t matter which is which because multiplication is
commutative.)
(2x + b)(x + d)
The numbers b and d must multiply up to 1, and so one of them must be 1 andthe other must be 1.
(2x + 1)(x + 1)
[email protected] Factorising 22 / 43
Harder Factorisation: Example
Factorise 2x2 + 3x + 1.
We need to write this in the form (ax + b)(cx + d).
The numbers a and c must multiply up to 2, and so one of them must be 1 andthe other must be 2. (It doesn’t matter which is which because multiplication is
commutative.)
(2x + b)(x + d)
The numbers b and d must multiply up to 1, and so one of them must be 1 andthe other must be 1.
(2x + 1)(x + 1)
[email protected] Factorising 22 / 43
Harder Factorisation: Example
Factorise 2x2 + 3x + 1.
We need to write this in the form (ax + b)(cx + d).
The numbers a and c must multiply up to 2, and so one of them must be 1 andthe other must be 2.
(It doesn’t matter which is which because multiplication iscommutative.)
(2x + b)(x + d)
The numbers b and d must multiply up to 1, and so one of them must be 1 andthe other must be 1.
(2x + 1)(x + 1)
[email protected] Factorising 22 / 43
Harder Factorisation: Example
Factorise 2x2 + 3x + 1.
We need to write this in the form (ax + b)(cx + d).
The numbers a and c must multiply up to 2, and so one of them must be 1 andthe other must be 2. (It doesn’t matter which is which because multiplication is
commutative.)
(2x + b)(x + d)
The numbers b and d must multiply up to 1, and so one of them must be 1 andthe other must be 1.
(2x + 1)(x + 1)
[email protected] Factorising 22 / 43
Harder Factorisation: Example
Factorise 2x2 + 3x + 1.
We need to write this in the form (ax + b)(cx + d).
The numbers a and c must multiply up to 2, and so one of them must be 1 andthe other must be 2. (It doesn’t matter which is which because multiplication is
commutative.)
(2x + b)(x + d)
The numbers b and d must multiply up to 1, and so one of them must be 1 andthe other must be 1.
(2x + 1)(x + 1)
[email protected] Factorising 22 / 43
Harder Factorisation: Example
Factorise 7x2 + 15x + 2.
We need to write this in the form (ax + b)(cx + d).
The numbers a and c must multiply to 7, and so one of them must be 1 and theother must be 7. It doesn’t matter which is which.
(7x + b)(x + d)
The numbers b and d must multiply to 2, so one of them must be 1 and the othermust be 2. The question is, which one is which? There are two possibilities:
(7x + 2)(x + 1) (7x + 1)(x + 2)
To decide which one is correct, expand them both:
7x2 + 9x + 2 7x2 + 15x + 2
[email protected] Factorising 23 / 43
Harder Factorisation: Example
Factorise 7x2 + 15x + 2.
We need to write this in the form (ax + b)(cx + d).
The numbers a and c must multiply to 7, and so one of them must be 1 and theother must be 7. It doesn’t matter which is which.
(7x + b)(x + d)
The numbers b and d must multiply to 2, so one of them must be 1 and the othermust be 2. The question is, which one is which? There are two possibilities:
(7x + 2)(x + 1) (7x + 1)(x + 2)
To decide which one is correct, expand them both:
7x2 + 9x + 2 7x2 + 15x + 2
[email protected] Factorising 23 / 43
Harder Factorisation: Example
Factorise 7x2 + 15x + 2.
We need to write this in the form (ax + b)(cx + d).
The numbers a and c must multiply to 7, and so one of them must be 1 and theother must be 7. It doesn’t matter which is which.
(7x + b)(x + d)
The numbers b and d must multiply to 2, so one of them must be 1 and the othermust be 2. The question is, which one is which? There are two possibilities:
(7x + 2)(x + 1) (7x + 1)(x + 2)
To decide which one is correct, expand them both:
7x2 + 9x + 2 7x2 + 15x + 2
[email protected] Factorising 23 / 43
Harder Factorisation: Example
Factorise 7x2 + 15x + 2.
We need to write this in the form (ax + b)(cx + d).
The numbers a and c must multiply to 7, and so one of them must be 1 and theother must be 7. It doesn’t matter which is which.
(7x + b)(x + d)
The numbers b and d must multiply to 2, so one of them must be 1 and the othermust be 2. The question is, which one is which? There are two possibilities:
(7x + 2)(x + 1) (7x + 1)(x + 2)
To decide which one is correct, expand them both:
7x2 + 9x + 2 7x2 + 15x + 2
[email protected] Factorising 23 / 43
Harder Factorisation: Example
Factorise 7x2 + 15x + 2.
We need to write this in the form (ax + b)(cx + d).
The numbers a and c must multiply to 7, and so one of them must be 1 and theother must be 7. It doesn’t matter which is which.
(7x + b)(x + d)
The numbers b and d must multiply to 2, so one of them must be 1 and the othermust be 2. The question is, which one is which? There are two possibilities:
(7x + 2)(x + 1) (7x + 1)(x + 2)
To decide which one is correct, expand them both:
7x2 + 9x + 2 7x2 + 15x + 2
[email protected] Factorising 23 / 43
Harder Factorisation: Example
Factorise 7x2 + 15x + 2.
We need to write this in the form (ax + b)(cx + d).
The numbers a and c must multiply to 7, and so one of them must be 1 and theother must be 7. It doesn’t matter which is which.
(7x + b)(x + d)
The numbers b and d must multiply to 2, so one of them must be 1 and the othermust be 2. The question is, which one is which? There are two possibilities:
(7x + 2)(x + 1) (7x + 1)(x + 2)
To decide which one is correct, expand them both:
7x2 + 9x + 2
7x2 + 15x + 2
[email protected] Factorising 23 / 43
Harder Factorisation: Example
Factorise 7x2 + 15x + 2.
We need to write this in the form (ax + b)(cx + d).
The numbers a and c must multiply to 7, and so one of them must be 1 and theother must be 7. It doesn’t matter which is which.
(7x + b)(x + d)
The numbers b and d must multiply to 2, so one of them must be 1 and the othermust be 2. The question is, which one is which? There are two possibilities:
(7x + 2)(x + 1) (7x + 1)(x + 2)
To decide which one is correct, expand them both:
7x2 + 9x + 2 7x2 + 15x + 2
[email protected] Factorising 23 / 43
Harder Factorisation: Example
Factorise 7x2 + 15x + 2.
We need to write this in the form (ax + b)(cx + d).
The numbers a and c must multiply to 7, and so one of them must be 1 and theother must be 7. It doesn’t matter which is which.
(7x + b)(x + d)
The numbers b and d must multiply to 2, so one of them must be 1 and the othermust be 2. The question is, which one is which? There are two possibilities:
(7x + 2)(x + 1) (7x + 1)(x + 2)
To decide which one is correct, expand them both:
7x2 + 9x + 2 7x2 + 15x + 2
[email protected] Factorising 24 / 43
Harder Factorisation: Example
Factorise 6x2 − 11x − 10.
We need to write this in the form (ax + b)(cx + d).
The problem now is that both 6 and 10 have multiple possible factorisations!There are in fact 16 potential answers to test.
The best way to navigate through these options is with a combination of educatedguessing and trial-and-error. Start by writing out the potential factorisations of 6
and 10 as follows:
3
2
5
2
10
1
-
-
���
���*HHHHHHj-
-
In this case, the number term (−10) is negative so we need to get a pair whosedifference is 11.
6=
[email protected] Factorising 25 / 43
Harder Factorisation: Example
Factorise 6x2 − 11x − 10.
We need to write this in the form (ax + b)(cx + d).
The problem now is that both 6 and 10 have multiple possible factorisations!There are in fact 16 potential answers to test.
The best way to navigate through these options is with a combination of educatedguessing and trial-and-error. Start by writing out the potential factorisations of 6
and 10 as follows:
3
2
5
2
10
1
-
-
���
���*HHHHHHj-
-
In this case, the number term (−10) is negative so we need to get a pair whosedifference is 11.
6=
[email protected] Factorising 25 / 43
Harder Factorisation: Example
Factorise 6x2 − 11x − 10.
We need to write this in the form (ax + b)(cx + d).
The problem now is that both 6 and 10 have multiple possible factorisations!There are in fact 16 potential answers to test.
The best way to navigate through these options is with a combination of educatedguessing and trial-and-error. Start by writing out the potential factorisations of 6
and 10 as follows:
3
2
5
2
10
1
-
-
���
���*HHHHHHj-
-
In this case, the number term (−10) is negative so we need to get a pair whosedifference is 11.
6=
[email protected] Factorising 25 / 43
Harder Factorisation: Example
Factorise 6x2 − 11x − 10.
We need to write this in the form (ax + b)(cx + d).
The problem now is that both 6 and 10 have multiple possible factorisations!There are in fact 16 potential answers to test.
The best way to navigate through these options is with a combination of educatedguessing and trial-and-error.
Start by writing out the potential factorisations of 6and 10 as follows:
3
2
5
2
10
1
-
-
���
���*HHHHHHj-
-
In this case, the number term (−10) is negative so we need to get a pair whosedifference is 11.
6=
[email protected] Factorising 25 / 43
Harder Factorisation: Example
Factorise 6x2 − 11x − 10.
We need to write this in the form (ax + b)(cx + d).
The problem now is that both 6 and 10 have multiple possible factorisations!There are in fact 16 potential answers to test.
The best way to navigate through these options is with a combination of educatedguessing and trial-and-error. Start by writing out the potential factorisations of 6
and 10 as follows:
3
2
5
2
10
1
-
-
���
���*HHHHHHj-
-
In this case, the number term (−10) is negative so we need to get a pair whosedifference is 11.
6=
[email protected] Factorising 25 / 43
Harder Factorisation: Example
Factorise 6x2 − 11x − 10.
We need to write this in the form (ax + b)(cx + d).
The problem now is that both 6 and 10 have multiple possible factorisations!There are in fact 16 potential answers to test.
The best way to navigate through these options is with a combination of educatedguessing and trial-and-error. Start by writing out the potential factorisations of 6
and 10 as follows:
3
2
6
1
5
2
10
1
-
-
���
���*HHHHHHj-
-
In this case, the number term (−10) is negative so we need to get a pair whosedifference is 11.
[email protected] Factorising 25 / 43
Harder Factorisation: Example
Factorise 6x2 − 11x − 10.
We need to write this in the form (ax + b)(cx + d).
The problem now is that both 6 and 10 have multiple possible factorisations!There are in fact 16 potential answers to test.
The best way to navigate through these options is with a combination of educatedguessing and trial-and-error. Start by writing out the potential factorisations of 6
and 10 as follows:
3
2
6
1
5
2
10
1
-
-
���
���*HHHHHHj-
-
In this case, the number term (−10) is negative so we need to get a pair whosedifference is 11.
[email protected] Factorising 25 / 43
Harder Factorisation: Example
Factorise 6x2 − 11x − 10.
We need to write this in the form (ax + b)(cx + d).
The problem now is that both 6 and 10 have multiple possible factorisations!There are in fact 16 potential answers to test.
The best way to navigate through these options is with a combination of educatedguessing and trial-and-error. Start by writing out the potential factorisations of 6
and 10 as follows:
3
2
6
1
5
2
10
1
-
-
���
���*HHHHHHj-
-
In this case, the number term (−10) is negative so we need to get a pair whosedifference is 11.
1× 2 = 2 and 6× 5 = 30. Difference 6= 11. Try [email protected] Factorising 25 / 43
Harder Factorisation: Example
Factorise 6x2 − 11x − 10.
We need to write this in the form (ax + b)(cx + d).
The problem now is that both 6 and 10 have multiple possible factorisations!There are in fact 16 potential answers to test.
The best way to navigate through these options is with a combination of educatedguessing and trial-and-error. Start by writing out the potential factorisations of 6
and 10 as follows:
3
2
6
1
5
2
10
1
-
-
���
���*HHHHHHj
-
-
In this case, the number term (−10) is negative so we need to get a pair whosedifference is 11.
1× 5 = 5 and 6× 2 = 12. Difference 6= 11. Try [email protected] Factorising 25 / 43
Harder Factorisation: Example
Factorise 6x2 − 11x − 10.
We need to write this in the form (ax + b)(cx + d).
The problem now is that both 6 and 10 have multiple possible factorisations!There are in fact 16 potential answers to test.
The best way to navigate through these options is with a combination of educatedguessing and trial-and-error. Start by writing out the potential factorisations of 6
and 10 as follows:
3
2
6
1
5
2
10
1
-
-
���
���*HHHHHHj
-
-
In this case, the number term (−10) is negative so we need to get a pair whosedifference is 11.
2× 2 = 4 and 3× 5 = 15. Difference = 11. We have a [email protected] Factorising 25 / 43
Harder Factorisation: Example
Factorise 6x2 − 11x − 10.
We need to write this in the form (ax + b)(cx + d).
We have found that 2× 2 = 4 and 3× 5 = 15.
(2x5)(3x2)
All we need to do now is place the “+” and “−” signs in the appropriate brackets.
So our answer is
6x2 − 11x − 10 = (2x − 5)(3x + 2).
[email protected] Factorising 26 / 43
Harder Factorisation: Example
Factorise 6x2 − 11x − 10.
We need to write this in the form (ax + b)(cx + d).
We have found that 2× 2 = 4 and 3× 5 = 15.
(2x 5)(3x 2)
All we need to do now is place the “+” and “−” signs in the appropriate brackets.
So our answer is
6x2 − 11x − 10 = (2x − 5)(3x + 2).
[email protected] Factorising 26 / 43
Harder Factorisation: Example
Factorise 6x2 − 11x − 10.
We need to write this in the form (ax + b)(cx + d).
We have found that 2× 2 = 4 and 3× 5 = 15.
(2x 5)(3x 2)
All we need to do now is place the “+” and “−” signs in the appropriate brackets.
So our answer is
6x2 − 11x − 10 = (2x − 5)(3x + 2).
[email protected] Factorising 26 / 43
Harder Factorisation: Example
Factorise 6x2 − 11x − 10.
We need to write this in the form (ax + b)(cx + d).
We have found that 2× 2 = 4 and 3× 5 = 15.
(2x − 5)(3x + 2)
All we need to do now is place the “+” and “−” signs in the appropriate brackets.
So our answer is
6x2 − 11x − 10 = (2x − 5)(3x + 2).
[email protected] Factorising 26 / 43
Harder Factorisation: Example
Factorise 6x2 − 11x − 10.
We need to write this in the form (ax + b)(cx + d).
We have found that 2× 2 = 4 and 3× 5 = 15.
(2x − 5)(3x + 2)
All we need to do now is place the “+” and “−” signs in the appropriate brackets.
So our answer is
6x2 − 11x − 10 = (2x − 5)(3x + 2).
[email protected] Factorising 26 / 43
A final note on double bracket factorisations
Note: Not all expressions of the form ax2 + bx + c can be factorised into twobrackets easily.
Some can’t be facorised at all. For example
x2 + 2x + 2
looks easy but cannot be factorised.
[email protected] Factorising 27 / 43
A final note on double bracket factorisations
Note: Not all expressions of the form ax2 + bx + c can be factorised into twobrackets easily. Some can’t be facorised at all. For example
x2 + 2x + 2
looks easy but cannot be factorised.
[email protected] Factorising 27 / 43
Why is factorising useful?
Factorising is handy for simplifying expressions and equations, which makesformulas more efficient to use and problems easier to solve.
Solve for x : x2 = 6x .
x2 − 6x = 0 (get x terms together)
x(x − 6) = 0 (factorise)
We know that 0× a = 0, whatever a is, so only one of the two factors aboveneeds to be 0 to solve the equation:
Either x = 0 or x − 6 = 0.
Hence, the solutions are x = 0 or 6.
[email protected] Factorising 28 / 43
Why is factorising useful?
Factorising is handy for simplifying expressions and equations, which makesformulas more efficient to use and problems easier to solve.
Solve for x : x2 = 6x .
x2 − 6x = 0 (get x terms together)
x(x − 6) = 0 (factorise)
We know that 0× a = 0, whatever a is, so only one of the two factors aboveneeds to be 0 to solve the equation:
Either x = 0 or x − 6 = 0.
Hence, the solutions are x = 0 or 6.
[email protected] Factorising 28 / 43
Why is factorising useful?
Factorising is handy for simplifying expressions and equations, which makesformulas more efficient to use and problems easier to solve.
Solve for x : x2 = 6x .
x2 − 6x = 0 (get x terms together)
x(x − 6) = 0 (factorise)
We know that 0× a = 0, whatever a is, so only one of the two factors aboveneeds to be 0 to solve the equation:
Either x = 0 or x − 6 = 0.
Hence, the solutions are x = 0 or 6.
[email protected] Factorising 28 / 43
Why is factorising useful?
Factorising is handy for simplifying expressions and equations, which makesformulas more efficient to use and problems easier to solve.
Solve for x : x2 = 6x .
x2 − 6x = 0 (get x terms together)
x(x − 6) = 0 (factorise)
We know that 0× a = 0, whatever a is, so only one of the two factors aboveneeds to be 0 to solve the equation:
Either x = 0 or x − 6 = 0.
Hence, the solutions are x = 0 or 6.
[email protected] Factorising 28 / 43
Why is factorising useful?
Factorising is handy for simplifying expressions and equations, which makesformulas more efficient to use and problems easier to solve.
Solve for x : x2 = 6x .
x2 − 6x = 0 (get x terms together)
x(x − 6) = 0 (factorise)
We know that 0× a = 0, whatever a is, so only one of the two factors aboveneeds to be 0 to solve the equation:
Either x = 0 or x − 6 = 0.
Hence, the solutions are x = 0 or 6.
[email protected] Factorising 28 / 43
Why is factorising useful?
Factorising is handy for simplifying expressions and equations, which makesformulas more efficient to use and problems easier to solve.
Solve for x : x2 = 6x .
x2 − 6x = 0 (get x terms together)
x(x − 6) = 0 (factorise)
We know that 0× a = 0, whatever a is, so only one of the two factors aboveneeds to be 0 to solve the equation:
Either x = 0 or x − 6 = 0.
Hence, the solutions are x = 0 or 6.
[email protected] Factorising 28 / 43
Why is factorising useful?
Factorising is handy for simplifying expressions and equations, which makesformulas more efficient to use and problems easier to solve.
Solve for x : x2 = 6x .
x2 − 6x = 0 (get x terms together)
x(x − 6) = 0 (factorise)
We know that 0× a = 0, whatever a is, so only one of the two factors aboveneeds to be 0 to solve the equation:
Either x = 0 or x − 6 = 0.
Hence, the solutions are x = 0 or 6.
[email protected] Factorising 28 / 43
Why is factorising useful?
Factorising can also simplify algebraic fractions:
4x + 6
2=
2(2x + 3)
2(factorise)
=1�2(2x + 3)
�21(cancel common factors)
= 2x + 3
[email protected] Factorising 29 / 43
Why is factorising useful?
Factorising can also simplify algebraic fractions:
4x + 6
2=
2(2x + 3)
2(factorise)
=1�2(2x + 3)
�21(cancel common factors)
= 2x + 3
[email protected] Factorising 29 / 43
Why is factorising useful?
Factorising can also simplify algebraic fractions:
4x + 6
2=
2(2x + 3)
2(factorise)
=1�2(2x + 3)
�21(cancel common factors)
= 2x + 3
[email protected] Factorising 29 / 43
Why is factorising useful?
Factorising can also simplify algebraic fractions:
4x + 6
2=
2(2x + 3)
2(factorise)
=1�2(2x + 3)
�21(cancel common factors)
= 2x + 3
[email protected] Factorising 29 / 43
Why is factorising useful?
Factorising can also simplify algebraic fractions:
4x + 6
2=
2(2x + 3)
2(factorise)
=1�2(2x + 3)
�21(cancel common factors)
= 2x + 3
[email protected] Factorising 29 / 43
Why is factorising useful?
Factorising can also simplify algebraic fractions:
x2 + 3x + 2
x + 1=
(x + 1)(x + 2)
x + 1(factorise)
=1��
��(x + 1)(x + 2)
���x + 11(cancel common factors)
= x + 2
Note: Watch out when you cancel terms involving variables. The original fractionmakes it clear that there is a problem when x = −1 because the fraction becomes
00 , which is an indeterminate quantity. This problem is no longer obvious in the
simplified version. Usually, we would write the answer as
x + 2, x 6= −1
[email protected] Factorising 30 / 43
Why is factorising useful?
Factorising can also simplify algebraic fractions:
x2 + 3x + 2
x + 1=
(x + 1)(x + 2)
x + 1(factorise)
=1��
��(x + 1)(x + 2)
���x + 11(cancel common factors)
= x + 2
Note: Watch out when you cancel terms involving variables. The original fractionmakes it clear that there is a problem when x = −1 because the fraction becomes
00 , which is an indeterminate quantity. This problem is no longer obvious in the
simplified version. Usually, we would write the answer as
x + 2, x 6= −1
[email protected] Factorising 30 / 43
Why is factorising useful?
Factorising can also simplify algebraic fractions:
x2 + 3x + 2
x + 1=
(x + 1)(x + 2)
x + 1(factorise)
=1��
��(x + 1)(x + 2)
���x + 11(cancel common factors)
= x + 2
Note: Watch out when you cancel terms involving variables. The original fractionmakes it clear that there is a problem when x = −1 because the fraction becomes
00 , which is an indeterminate quantity. This problem is no longer obvious in the
simplified version. Usually, we would write the answer as
x + 2, x 6= −1
[email protected] Factorising 30 / 43
Why is factorising useful?
Factorising can also simplify algebraic fractions:
x2 + 3x + 2
x + 1=
(x + 1)(x + 2)
x + 1(factorise)
=1��
��(x + 1)(x + 2)
���x + 11(cancel common factors)
= x + 2
Note: Watch out when you cancel terms involving variables. The original fractionmakes it clear that there is a problem when x = −1 because the fraction becomes
00 , which is an indeterminate quantity. This problem is no longer obvious in the
simplified version. Usually, we would write the answer as
x + 2, x 6= −1
[email protected] Factorising 30 / 43
Why is factorising useful?
Factorising can also simplify algebraic fractions:
x2 + 3x + 2
x + 1=
(x + 1)(x + 2)
x + 1(factorise)
=1��
��(x + 1)(x + 2)
���x + 11(cancel common factors)
= x + 2
Note: Watch out when you cancel terms involving variables. The original fractionmakes it clear that there is a problem when x = −1 because the fraction becomes
00 , which is an indeterminate quantity. This problem is no longer obvious in the
simplified version. Usually, we would write the answer as
x + 2, x 6= −1
[email protected] Factorising 30 / 43
Why is factorising useful?
Factorising can also simplify algebraic fractions:
x2 + 3x + 2
x + 1=
(x + 1)(x + 2)
x + 1(factorise)
=1��
��(x + 1)(x + 2)
���x + 11(cancel common factors)
= x + 2
Note: Watch out when you cancel terms involving variables. The original fractionmakes it clear that there is a problem when x = −1 because the fraction becomes
00 , which is an indeterminate quantity. This problem is no longer obvious in the
simplified version. Usually, we would write the answer as
x + 2, x 6= −1
[email protected] Factorising 30 / 43
Polynomials
This final section is about polynomials, a topic which may not be on your mathssyllabus.
[email protected] Factorising 31 / 43
Polynomials
A polynomial is a bunch of terms involving whole number powers (ie. positiveintegers) of a variable added/subtracted together.
We might also have a constant term (a single number) as well.
3x2 − 6x + 7x5 + 2
Look at the above polynomial. Every term in it is either a positive integer powerof the variable x or a constant term.
Term Power of x
3x2 27x5 5−6x 1
2 constant term or “2x0”(so the power of x is 0)
[email protected] Factorising 32 / 43
Polynomials
A polynomial is a bunch of terms involving whole number powers (ie. positiveintegers) of a variable added/subtracted together.
We might also have a constant term (a single number) as well.
3x2 − 6x + 7x5 + 2
Look at the above polynomial. Every term in it is either a positive integer powerof the variable x or a constant term.
Term Power of x
3x2 27x5 5−6x 1
2 constant term or “2x0”(so the power of x is 0)
[email protected] Factorising 32 / 43
Polynomials
A polynomial is a bunch of terms involving whole number powers (ie. positiveintegers) of a variable added/subtracted together.
We might also have a constant term (a single number) as well.
3x2 − 6x + 7x5 + 2
Look at the above polynomial. Every term in it is either a positive integer powerof the variable x or a constant term.
Term Power of x
3x2 27x5 5−6x 1
2 constant term or “2x0”(so the power of x is 0)
[email protected] Factorising 32 / 43
Polynomials
A polynomial is a bunch of terms involving whole number powers (ie. positiveintegers) of a variable added/subtracted together.
We might also have a constant term (a single number) as well.
3x2 − 6x + 7x5 + 2
Look at the above polynomial. Every term in it is either a positive integer powerof the variable x or a constant term.
Term Power of x
3x2 27x5 5−6x 1
2 constant term or “2x0”(so the power of x is 0)
[email protected] Factorising 32 / 43
Polynomials
A polynomial is a bunch of terms involving whole number powers (ie. positiveintegers) of a variable added/subtracted together.
We might also have a constant term (a single number) as well.
3x2 − 6x + 7x5 + 2
Look at the above polynomial. Every term in it is either a positive integer powerof the variable x or a constant term.
Term Power of x
3x2 27x5 5−6x 1
2 constant term or “2x0”(so the power of x is 0)
[email protected] Factorising 32 / 43
Polynomials
A polynomial is a bunch of terms involving whole number powers (ie. positiveintegers) of a variable added/subtracted together.
We might also have a constant term (a single number) as well.
3x2 − 6x + 7x5 + 2
Look at the above polynomial. Every term in it is either a positive integer powerof the variable x or a constant term.
Term Power of x
3x2
27x5 5−6x 1
2 constant term or “2x0”(so the power of x is 0)
[email protected] Factorising 32 / 43
Polynomials
A polynomial is a bunch of terms involving whole number powers (ie. positiveintegers) of a variable added/subtracted together.
We might also have a constant term (a single number) as well.
3x2 − 6x + 7x5 + 2
Look at the above polynomial. Every term in it is either a positive integer powerof the variable x or a constant term.
Term Power of x
3x2 2
7x5 5−6x 1
2 constant term or “2x0”(so the power of x is 0)
[email protected] Factorising 32 / 43
Polynomials
A polynomial is a bunch of terms involving whole number powers (ie. positiveintegers) of a variable added/subtracted together.
We might also have a constant term (a single number) as well.
3x2 − 6x + 7x5 + 2
Look at the above polynomial. Every term in it is either a positive integer powerof the variable x or a constant term.
Term Power of x
3x2 27x5
5−6x 1
2 constant term or “2x0”(so the power of x is 0)
[email protected] Factorising 32 / 43
Polynomials
A polynomial is a bunch of terms involving whole number powers (ie. positiveintegers) of a variable added/subtracted together.
We might also have a constant term (a single number) as well.
3x2 − 6x + 7x5 + 2
Look at the above polynomial. Every term in it is either a positive integer powerof the variable x or a constant term.
Term Power of x
3x2 27x5 5
−6x 12 constant term or “2x0”
(so the power of x is 0)
[email protected] Factorising 32 / 43
Polynomials
A polynomial is a bunch of terms involving whole number powers (ie. positiveintegers) of a variable added/subtracted together.
We might also have a constant term (a single number) as well.
3x2 − 6x + 7x5 + 2
Look at the above polynomial. Every term in it is either a positive integer powerof the variable x or a constant term.
Term Power of x
3x2 27x5 5−6x
12 constant term or “2x0”
(so the power of x is 0)
[email protected] Factorising 32 / 43
Polynomials
A polynomial is a bunch of terms involving whole number powers (ie. positiveintegers) of a variable added/subtracted together.
We might also have a constant term (a single number) as well.
3x2 − 6x + 7x5 + 2
Look at the above polynomial. Every term in it is either a positive integer powerof the variable x or a constant term.
Term Power of x
3x2 27x5 5−6x 1
2 constant term or “2x0”(so the power of x is 0)
[email protected] Factorising 32 / 43
Polynomials
A polynomial is a bunch of terms involving whole number powers (ie. positiveintegers) of a variable added/subtracted together.
We might also have a constant term (a single number) as well.
3x2 − 6x + 7x5 + 2
Look at the above polynomial. Every term in it is either a positive integer powerof the variable x or a constant term.
Term Power of x
3x2 27x5 5−6x 1
2
constant term or “2x0”(so the power of x is 0)
[email protected] Factorising 32 / 43
Polynomials
A polynomial is a bunch of terms involving whole number powers (ie. positiveintegers) of a variable added/subtracted together.
We might also have a constant term (a single number) as well.
3x2 − 6x + 7x5 + 2
Look at the above polynomial. Every term in it is either a positive integer powerof the variable x or a constant term.
Term Power of x
3x2 27x5 5−6x 1
2 constant term
or “2x0”(so the power of x is 0)
[email protected] Factorising 32 / 43
Polynomials
A polynomial is a bunch of terms involving whole number powers (ie. positiveintegers) of a variable added/subtracted together.
We might also have a constant term (a single number) as well.
3x2 − 6x + 7x5 + 2
Look at the above polynomial. Every term in it is either a positive integer powerof the variable x or a constant term.
Term Power of x
3x2 27x5 5−6x 1
2 constant term or “2x0”
(so the power of x is 0)
[email protected] Factorising 32 / 43
Polynomials
A polynomial is a bunch of terms involving whole number powers (ie. positiveintegers) of a variable added/subtracted together.
We might also have a constant term (a single number) as well.
3x2 − 6x + 7x5 + 2
Look at the above polynomial. Every term in it is either a positive integer powerof the variable x or a constant term.
Term Power of x
3x2 27x5 5−6x 1
2 constant term or “2x0”(so the power of x is 0)
[email protected] Factorising 32 / 43
Polynomials: Examples
7p8 − 62p2 + 5− p3
7y2 − 3y4
3 + x
23q
5 + q4 + 3q7 − 2q + 1
(Note that the numbers multiplying the variable parts do not have to be wholenumbers.)
[email protected] Factorising 33 / 43
Polynomials: Examples
7p8 − 62p2 + 5− p3
7y2 − 3y4
3 + x
23q
5 + q4 + 3q7 − 2q + 1
(Note that the numbers multiplying the variable parts do not have to be wholenumbers.)
[email protected] Factorising 33 / 43
Polynomials: Examples
7p8 − 62p2 + 5− p3
7y2 − 3y4
3 + x
23q
5 + q4 + 3q7 − 2q + 1
(Note that the numbers multiplying the variable parts do not have to be wholenumbers.)
[email protected] Factorising 33 / 43
Polynomials: Examples
7p8 − 62p2 + 5− p3
7y2 − 3y4
3 + x
23q
5 + q4 + 3q7 − 2q + 1
(Note that the numbers multiplying the variable parts do not have to be wholenumbers.)
[email protected] Factorising 33 / 43
Polynomials: A Convention
Polynomials are usually written with their powers in descending order.
7p8 − 62p2 + 5− p3
−→ 7p8 − p3 − 62p2 + 5
7y2 − 3y4 −→ −3y4 + 7y2
3 + x −→ x + 3
23q
5 + q4 + 3q7 − 2q + 1 −→ 3q7 + 23q
5 + q4 − 2q + 1
[email protected] Factorising 34 / 43
Polynomials: A Convention
Polynomials are usually written with their powers in descending order.
7p8 − 62p2 + 5− p3 −→ 7p8 − p3 − 62p2 + 5
7y2 − 3y4 −→ −3y4 + 7y2
3 + x −→ x + 3
23q
5 + q4 + 3q7 − 2q + 1 −→ 3q7 + 23q
5 + q4 − 2q + 1
[email protected] Factorising 34 / 43
Polynomials: A Convention
Polynomials are usually written with their powers in descending order.
7p8 − 62p2 + 5− p3 −→ 7p8 − p3 − 62p2 + 5
7y2 − 3y4
−→ −3y4 + 7y2
3 + x −→ x + 3
23q
5 + q4 + 3q7 − 2q + 1 −→ 3q7 + 23q
5 + q4 − 2q + 1
[email protected] Factorising 34 / 43
Polynomials: A Convention
Polynomials are usually written with their powers in descending order.
7p8 − 62p2 + 5− p3 −→ 7p8 − p3 − 62p2 + 5
7y2 − 3y4 −→ −3y4 + 7y2
3 + x −→ x + 3
23q
5 + q4 + 3q7 − 2q + 1 −→ 3q7 + 23q
5 + q4 − 2q + 1
[email protected] Factorising 34 / 43
Polynomials: A Convention
Polynomials are usually written with their powers in descending order.
7p8 − 62p2 + 5− p3 −→ 7p8 − p3 − 62p2 + 5
7y2 − 3y4 −→ −3y4 + 7y2
3 + x
−→ x + 3
23q
5 + q4 + 3q7 − 2q + 1 −→ 3q7 + 23q
5 + q4 − 2q + 1
[email protected] Factorising 34 / 43
Polynomials: A Convention
Polynomials are usually written with their powers in descending order.
7p8 − 62p2 + 5− p3 −→ 7p8 − p3 − 62p2 + 5
7y2 − 3y4 −→ −3y4 + 7y2
3 + x −→ x + 3
23q
5 + q4 + 3q7 − 2q + 1 −→ 3q7 + 23q
5 + q4 − 2q + 1
[email protected] Factorising 34 / 43
Polynomials: A Convention
Polynomials are usually written with their powers in descending order.
7p8 − 62p2 + 5− p3 −→ 7p8 − p3 − 62p2 + 5
7y2 − 3y4 −→ −3y4 + 7y2
3 + x −→ x + 3
23q
5 + q4 + 3q7 − 2q + 1
−→ 3q7 + 23q
5 + q4 − 2q + 1
[email protected] Factorising 34 / 43
Polynomials: A Convention
Polynomials are usually written with their powers in descending order.
7p8 − 62p2 + 5− p3 −→ 7p8 − p3 − 62p2 + 5
7y2 − 3y4 −→ −3y4 + 7y2
3 + x −→ x + 3
23q
5 + q4 + 3q7 − 2q + 1 −→ 3q7 + 23q
5 + q4 − 2q + 1
[email protected] Factorising 34 / 43
Polynomials: Terminology
Consider the following polynomial.
3x7 − 4x5 + 37x
4 − 2x + 9
We say it is a “polynomial in the variable x”.
The order of a polynomial is the highest power of x present. In this case, theorder is 7.
The coefficient of x7 is 3.(The number multiplying the x part.)
The coefficient of x5 is −4.
The coefficient of x4 is 37 .
The coefficient of x is −2.
The coefficient of x0 is 9 (constant term).
The coefficient of x2 is 0 (because it isn’t there!).
[email protected] Factorising 35 / 43
Polynomials: Terminology
Consider the following polynomial.
3x7 − 4x5 + 37x
4 − 2x + 9
We say it is a “polynomial in the variable x”.
The order of a polynomial is the highest power of x present. In this case, theorder is 7.
The coefficient of x7 is 3.(The number multiplying the x part.)
The coefficient of x5 is −4.
The coefficient of x4 is 37 .
The coefficient of x is −2.
The coefficient of x0 is 9 (constant term).
The coefficient of x2 is 0 (because it isn’t there!).
[email protected] Factorising 35 / 43
Polynomials: Terminology
Consider the following polynomial.
3x7 − 4x5 + 37x
4 − 2x + 9
We say it is a “polynomial in the variable x”.
The order of a polynomial is the highest power of x present. In this case, theorder is
7.
The coefficient of x7 is 3.(The number multiplying the x part.)
The coefficient of x5 is −4.
The coefficient of x4 is 37 .
The coefficient of x is −2.
The coefficient of x0 is 9 (constant term).
The coefficient of x2 is 0 (because it isn’t there!).
[email protected] Factorising 35 / 43
Polynomials: Terminology
Consider the following polynomial.
3x7 − 4x5 + 37x
4 − 2x + 9
We say it is a “polynomial in the variable x”.
The order of a polynomial is the highest power of x present. In this case, theorder is 7.
The coefficient of x7 is 3.(The number multiplying the x part.)
The coefficient of x5 is −4.
The coefficient of x4 is 37 .
The coefficient of x is −2.
The coefficient of x0 is 9 (constant term).
The coefficient of x2 is 0 (because it isn’t there!).
[email protected] Factorising 35 / 43
Polynomials: Terminology
Consider the following polynomial.
3x7 − 4x5 + 37x
4 − 2x + 9
We say it is a “polynomial in the variable x”.
The order of a polynomial is the highest power of x present. In this case, theorder is 7.
The coefficient of x7 is 3.(The number multiplying the x part.)
The coefficient of x5 is −4.
The coefficient of x4 is 37 .
The coefficient of x is −2.
The coefficient of x0 is 9 (constant term).
The coefficient of x2 is 0 (because it isn’t there!).
[email protected] Factorising 35 / 43
Polynomials: Terminology
Consider the following polynomial.
3x7 − 4x5 + 37x
4 − 2x + 9
We say it is a “polynomial in the variable x”.
The order of a polynomial is the highest power of x present. In this case, theorder is 7.
The coefficient of x7 is 3.(The number multiplying the x part.)
The coefficient of x5 is
−4.
The coefficient of x4 is 37 .
The coefficient of x is −2.
The coefficient of x0 is 9 (constant term).
The coefficient of x2 is 0 (because it isn’t there!).
[email protected] Factorising 35 / 43
Polynomials: Terminology
Consider the following polynomial.
3x7 − 4x5 + 37x
4 − 2x + 9
We say it is a “polynomial in the variable x”.
The order of a polynomial is the highest power of x present. In this case, theorder is 7.
The coefficient of x7 is 3.(The number multiplying the x part.)
The coefficient of x5 is −4.
The coefficient of x4 is 37 .
The coefficient of x is −2.
The coefficient of x0 is 9 (constant term).
The coefficient of x2 is 0 (because it isn’t there!).
[email protected] Factorising 35 / 43
Polynomials: Terminology
Consider the following polynomial.
3x7 − 4x5 + 37x
4 − 2x + 9
We say it is a “polynomial in the variable x”.
The order of a polynomial is the highest power of x present. In this case, theorder is 7.
The coefficient of x7 is 3.(The number multiplying the x part.)
The coefficient of x5 is −4.
The coefficient of x4 is
37 .
The coefficient of x is −2.
The coefficient of x0 is 9 (constant term).
The coefficient of x2 is 0 (because it isn’t there!).
[email protected] Factorising 35 / 43
Polynomials: Terminology
Consider the following polynomial.
3x7 − 4x5 + 37x
4 − 2x + 9
We say it is a “polynomial in the variable x”.
The order of a polynomial is the highest power of x present. In this case, theorder is 7.
The coefficient of x7 is 3.(The number multiplying the x part.)
The coefficient of x5 is −4.
The coefficient of x4 is 37 .
The coefficient of x is −2.
The coefficient of x0 is 9 (constant term).
The coefficient of x2 is 0 (because it isn’t there!).
[email protected] Factorising 35 / 43
Polynomials: Terminology
Consider the following polynomial.
3x7 − 4x5 + 37x
4 − 2x + 9
We say it is a “polynomial in the variable x”.
The order of a polynomial is the highest power of x present. In this case, theorder is 7.
The coefficient of x7 is 3.(The number multiplying the x part.)
The coefficient of x5 is −4.
The coefficient of x4 is 37 .
The coefficient of x is
−2.
The coefficient of x0 is 9 (constant term).
The coefficient of x2 is 0 (because it isn’t there!).
[email protected] Factorising 35 / 43
Polynomials: Terminology
Consider the following polynomial.
3x7 − 4x5 + 37x
4 − 2x + 9
We say it is a “polynomial in the variable x”.
The order of a polynomial is the highest power of x present. In this case, theorder is 7.
The coefficient of x7 is 3.(The number multiplying the x part.)
The coefficient of x5 is −4.
The coefficient of x4 is 37 .
The coefficient of x is −2.
The coefficient of x0 is 9 (constant term).
The coefficient of x2 is 0 (because it isn’t there!).
[email protected] Factorising 35 / 43
Polynomials: Terminology
Consider the following polynomial.
3x7 − 4x5 + 37x
4 − 2x + 9
We say it is a “polynomial in the variable x”.
The order of a polynomial is the highest power of x present. In this case, theorder is 7.
The coefficient of x7 is 3.(The number multiplying the x part.)
The coefficient of x5 is −4.
The coefficient of x4 is 37 .
The coefficient of x is −2.
The coefficient of x0 is
9 (constant term).
The coefficient of x2 is 0 (because it isn’t there!).
[email protected] Factorising 35 / 43
Polynomials: Terminology
Consider the following polynomial.
3x7 − 4x5 + 37x
4 − 2x + 9
We say it is a “polynomial in the variable x”.
The order of a polynomial is the highest power of x present. In this case, theorder is 7.
The coefficient of x7 is 3.(The number multiplying the x part.)
The coefficient of x5 is −4.
The coefficient of x4 is 37 .
The coefficient of x is −2.
The coefficient of x0 is 9 (constant term).
The coefficient of x2 is 0 (because it isn’t there!).
[email protected] Factorising 35 / 43
Polynomials: Terminology
Consider the following polynomial.
3x7 − 4x5 + 37x
4 − 2x + 9
We say it is a “polynomial in the variable x”.
The order of a polynomial is the highest power of x present. In this case, theorder is 7.
The coefficient of x7 is 3.(The number multiplying the x part.)
The coefficient of x5 is −4.
The coefficient of x4 is 37 .
The coefficient of x is −2.
The coefficient of x0 is 9 (constant term).
The coefficient of x2 is
0 (because it isn’t there!).
[email protected] Factorising 35 / 43
Polynomials: Terminology
Consider the following polynomial.
3x7 − 4x5 + 37x
4 − 2x + 9
We say it is a “polynomial in the variable x”.
The order of a polynomial is the highest power of x present. In this case, theorder is 7.
The coefficient of x7 is 3.(The number multiplying the x part.)
The coefficient of x5 is −4.
The coefficient of x4 is 37 .
The coefficient of x is −2.
The coefficient of x0 is 9 (constant term).
The coefficient of x2 is 0 (because it isn’t there!).
[email protected] Factorising 35 / 43
Polynomials: Terminology
Consider the following polynomial.
4x3 − 5x − 9
What is the order of this polynomial?
3
What is the coefficient of x3? 4
What is the coefficient of x2? 0
What is the coefficient of x? −5
What is the coeficient of x0? −9
[email protected] Factorising 36 / 43
Polynomials: Terminology
Consider the following polynomial.
4x3 − 5x − 9
What is the order of this polynomial? 3
What is the coefficient of x3? 4
What is the coefficient of x2? 0
What is the coefficient of x? −5
What is the coeficient of x0? −9
[email protected] Factorising 36 / 43
Polynomials: Terminology
Consider the following polynomial.
4x3 − 5x − 9
What is the order of this polynomial? 3
What is the coefficient of x3?
4
What is the coefficient of x2? 0
What is the coefficient of x? −5
What is the coeficient of x0? −9
[email protected] Factorising 36 / 43
Polynomials: Terminology
Consider the following polynomial.
4x3 − 5x − 9
What is the order of this polynomial? 3
What is the coefficient of x3? 4
What is the coefficient of x2? 0
What is the coefficient of x? −5
What is the coeficient of x0? −9
[email protected] Factorising 36 / 43
Polynomials: Terminology
Consider the following polynomial.
4x3 − 5x − 9
What is the order of this polynomial? 3
What is the coefficient of x3? 4
What is the coefficient of x2?
0
What is the coefficient of x? −5
What is the coeficient of x0? −9
[email protected] Factorising 36 / 43
Polynomials: Terminology
Consider the following polynomial.
4x3 − 5x − 9
What is the order of this polynomial? 3
What is the coefficient of x3? 4
What is the coefficient of x2? 0
What is the coefficient of x? −5
What is the coeficient of x0? −9
[email protected] Factorising 36 / 43
Polynomials: Terminology
Consider the following polynomial.
4x3 − 5x − 9
What is the order of this polynomial? 3
What is the coefficient of x3? 4
What is the coefficient of x2? 0
What is the coefficient of x?
−5
What is the coeficient of x0? −9
[email protected] Factorising 36 / 43
Polynomials: Terminology
Consider the following polynomial.
4x3 − 5x − 9
What is the order of this polynomial? 3
What is the coefficient of x3? 4
What is the coefficient of x2? 0
What is the coefficient of x? −5
What is the coeficient of x0? −9
[email protected] Factorising 36 / 43
Polynomials: Terminology
Consider the following polynomial.
4x3 − 5x − 9
What is the order of this polynomial? 3
What is the coefficient of x3? 4
What is the coefficient of x2? 0
What is the coefficient of x? −5
What is the coeficient of x0?
−9
[email protected] Factorising 36 / 43
Polynomials: Terminology
Consider the following polynomial.
4x3 − 5x − 9
What is the order of this polynomial? 3
What is the coefficient of x3? 4
What is the coefficient of x2? 0
What is the coefficient of x? −5
What is the coeficient of x0? −9
[email protected] Factorising 36 / 43
Importance
Why are polynomial expressions important?
They are easier to work with than other expressions (hard to know this unless youhave worked with other expressions).
Most other expressions can be very closely approximated by polynomialexpressions.
In a sense, polynomial expressions are mathematical “building blocks”.
[email protected] Factorising 37 / 43
Importance
Why are polynomial expressions important?
They are easier to work with than other expressions (hard to know this unless youhave worked with other expressions).
Most other expressions can be very closely approximated by polynomialexpressions.
In a sense, polynomial expressions are mathematical “building blocks”.
[email protected] Factorising 37 / 43
Importance
Why are polynomial expressions important?
They are easier to work with than other expressions (hard to know this unless youhave worked with other expressions).
Most other expressions can be very closely approximated by polynomialexpressions.
In a sense, polynomial expressions are mathematical “building blocks”.
[email protected] Factorising 37 / 43
Importance
Why are polynomial expressions important?
They are easier to work with than other expressions (hard to know this unless youhave worked with other expressions).
Most other expressions can be very closely approximated by polynomialexpressions.
In a sense, polynomial expressions are mathematical “building blocks”.
[email protected] Factorising 37 / 43
Adding and Subtracting Polynomials
When we add or subtract polynomials we get a new polynomial (just like we dowith numbers!). This can be done by adding and subtracting their like terms.
(4x2 + 3x + 7) + (2x2 + 5x + 2)
We may drop the brackets in this case (why?).
4x2 + 3x + 7 + 2x2 + 5x + 2
If you need to, shift the signed terms around so that like terms are next to eachother.
4x2 + 2x2 + 3x + 5x + 7 + 2
We now add like terms together.
6x2 + 8x + 9
[email protected] Factorising 38 / 43
Adding and Subtracting Polynomials
When we add or subtract polynomials we get a new polynomial (just like we dowith numbers!). This can be done by adding and subtracting their like terms.
(4x2 + 3x + 7) + (2x2 + 5x + 2)
We may drop the brackets in this case (why?).
4x2 + 3x + 7 + 2x2 + 5x + 2
If you need to, shift the signed terms around so that like terms are next to eachother.
4x2 + 2x2 + 3x + 5x + 7 + 2
We now add like terms together.
6x2 + 8x + 9
[email protected] Factorising 38 / 43
Adding and Subtracting Polynomials
When we add or subtract polynomials we get a new polynomial (just like we dowith numbers!). This can be done by adding and subtracting their like terms.
(4x2 + 3x + 7) + (2x2 + 5x + 2)
We may drop the brackets in this case (why?).
4x2 + 3x + 7 + 2x2 + 5x + 2
If you need to, shift the signed terms around so that like terms are next to eachother.
4x2 + 2x2 + 3x + 5x + 7 + 2
We now add like terms together.
6x2 + 8x + 9
[email protected] Factorising 38 / 43
Adding and Subtracting Polynomials
When we add or subtract polynomials we get a new polynomial (just like we dowith numbers!). This can be done by adding and subtracting their like terms.
(4x2 + 3x + 7) + (2x2 + 5x + 2)
We may drop the brackets in this case (why?).
4x2 + 3x + 7 + 2x2 + 5x + 2
If you need to, shift the signed terms around so that like terms are next to eachother.
4x2 + 2x2 + 3x + 5x + 7 + 2
We now add like terms together.
6x2 + 8x + 9
[email protected] Factorising 38 / 43
Adding and Subtracting Polynomials
Simplify the following:
(3x3 − 4x2 + 5)− (x3 + 3x − 4)
There is a negative sign to the left of the second set of brackets. This iseffectively a multiplier of −1. From the previous workshop, we know that this −1
gets distributed to each term in the bracket.
3x3 − 4x2 + 5− (x3 + 3x − 4)
= 3x3 − 4x2 + 5− x3− 3x + 4
(In short, every sign in the 2nd bracket changes.)
If you need to, shift the signed terms around so that like terms are next to eachother.
3x3 − x3 − 4x2 − 3x + 5 + 4
We now add like terms together.
2x3 − 4x2 − 3x + 9
[email protected] Factorising 39 / 43
Adding and Subtracting Polynomials
Simplify the following:
(3x3 − 4x2 + 5)− (x3 + 3x − 4)
There is a negative sign to the left of the second set of brackets. This iseffectively a multiplier of −1. From the previous workshop, we know that this −1
gets distributed to each term in the bracket.
3x3 − 4x2 + 5− (x3 + 3x − 4)
= 3x3 − 4x2 + 5− x3− 3x + 4
(In short, every sign in the 2nd bracket changes.)
If you need to, shift the signed terms around so that like terms are next to eachother.
3x3 − x3 − 4x2 − 3x + 5 + 4
We now add like terms together.
2x3 − 4x2 − 3x + 9
[email protected] Factorising 39 / 43
Adding and Subtracting Polynomials
Simplify the following:
(3x3 − 4x2 + 5)− (x3 + 3x − 4)
There is a negative sign to the left of the second set of brackets. This iseffectively a multiplier of −1. From the previous workshop, we know that this −1
gets distributed to each term in the bracket.
3x3 − 4x2 + 5− (x3 + 3x − 4)
= 3x3 − 4x2 + 5
− x3− 3x + 4
(In short, every sign in the 2nd bracket changes.)
If you need to, shift the signed terms around so that like terms are next to eachother.
3x3 − x3 − 4x2 − 3x + 5 + 4
We now add like terms together.
2x3 − 4x2 − 3x + 9
[email protected] Factorising 39 / 43
Adding and Subtracting Polynomials
Simplify the following:
(3x3 − 4x2 + 5)− (x3 + 3x − 4)
There is a negative sign to the left of the second set of brackets. This iseffectively a multiplier of −1. From the previous workshop, we know that this −1
gets distributed to each term in the bracket.
3x3 − 4x2 + 5− (x3 + 3x − 4)
= 3x3 − 4x2 + 5
− x3− 3x + 4
(In short, every sign in the 2nd bracket changes.)
If you need to, shift the signed terms around so that like terms are next to eachother.
3x3 − x3 − 4x2 − 3x + 5 + 4
We now add like terms together.
2x3 − 4x2 − 3x + 9
[email protected] Factorising 39 / 43
Adding and Subtracting Polynomials
Simplify the following:
(3x3 − 4x2 + 5)− (x3 + 3x − 4)
There is a negative sign to the left of the second set of brackets. This iseffectively a multiplier of −1. From the previous workshop, we know that this −1
gets distributed to each term in the bracket.
3x3 − 4x2 + 5− (x3 + 3x − 4)
= 3x3 − 4x2 + 5− x3
− 3x + 4
(In short, every sign in the 2nd bracket changes.)
If you need to, shift the signed terms around so that like terms are next to eachother.
3x3 − x3 − 4x2 − 3x + 5 + 4
We now add like terms together.
2x3 − 4x2 − 3x + 9
[email protected] Factorising 39 / 43
Adding and Subtracting Polynomials
Simplify the following:
(3x3 − 4x2 + 5)− (x3 + 3x − 4)
There is a negative sign to the left of the second set of brackets. This iseffectively a multiplier of −1. From the previous workshop, we know that this −1
gets distributed to each term in the bracket.
3x3 − 4x2 + 5− (x3 + 3x − 4)
= 3x3 − 4x2 + 5− x3
− 3x + 4
(In short, every sign in the 2nd bracket changes.)
If you need to, shift the signed terms around so that like terms are next to eachother.
3x3 − x3 − 4x2 − 3x + 5 + 4
We now add like terms together.
2x3 − 4x2 − 3x + 9
[email protected] Factorising 39 / 43
Adding and Subtracting Polynomials
Simplify the following:
(3x3 − 4x2 + 5)− (x3 + 3x − 4)
There is a negative sign to the left of the second set of brackets. This iseffectively a multiplier of −1. From the previous workshop, we know that this −1
gets distributed to each term in the bracket.
3x3 − 4x2 + 5− (x3 + 3x − 4)
= 3x3 − 4x2 + 5− x3− 3x
+ 4
(In short, every sign in the 2nd bracket changes.)
If you need to, shift the signed terms around so that like terms are next to eachother.
3x3 − x3 − 4x2 − 3x + 5 + 4
We now add like terms together.
2x3 − 4x2 − 3x + 9
[email protected] Factorising 39 / 43
Adding and Subtracting Polynomials
Simplify the following:
(3x3 − 4x2 + 5)− (x3 + 3x − 4)
There is a negative sign to the left of the second set of brackets. This iseffectively a multiplier of −1. From the previous workshop, we know that this −1
gets distributed to each term in the bracket.
3x3 − 4x2 + 5− (x3 + 3x − 4)
= 3x3 − 4x2 + 5− x3− 3x
+ 4
(In short, every sign in the 2nd bracket changes.)
If you need to, shift the signed terms around so that like terms are next to eachother.
3x3 − x3 − 4x2 − 3x + 5 + 4
We now add like terms together.
2x3 − 4x2 − 3x + 9
[email protected] Factorising 39 / 43
Adding and Subtracting Polynomials
Simplify the following:
(3x3 − 4x2 + 5)− (x3 + 3x − 4)
There is a negative sign to the left of the second set of brackets. This iseffectively a multiplier of −1. From the previous workshop, we know that this −1
gets distributed to each term in the bracket.
3x3 − 4x2 + 5− (x3 + 3x − 4)
= 3x3 − 4x2 + 5− x3− 3x + 4
(In short, every sign in the 2nd bracket changes.)
If you need to, shift the signed terms around so that like terms are next to eachother.
3x3 − x3 − 4x2 − 3x + 5 + 4
We now add like terms together.
2x3 − 4x2 − 3x + 9
[email protected] Factorising 39 / 43
Adding and Subtracting Polynomials
Simplify the following:
(3x3 − 4x2 + 5)− (x3 + 3x − 4)
There is a negative sign to the left of the second set of brackets. This iseffectively a multiplier of −1. From the previous workshop, we know that this −1
gets distributed to each term in the bracket.
3x3 − 4x2 + 5− (x3 + 3x − 4)
= 3x3 − 4x2 + 5− x3− 3x + 4
(In short, every sign in the 2nd bracket changes.)
If you need to, shift the signed terms around so that like terms are next to eachother.
3x3 − x3 − 4x2 − 3x + 5 + 4
We now add like terms together.
2x3 − 4x2 − 3x + 9
[email protected] Factorising 39 / 43
Adding and Subtracting Polynomials
Simplify the following:
(3x3 − 4x2 + 5)− (x3 + 3x − 4)
There is a negative sign to the left of the second set of brackets. This iseffectively a multiplier of −1. From the previous workshop, we know that this −1
gets distributed to each term in the bracket.
3x3 − 4x2 + 5− (x3 + 3x − 4)
= 3x3 − 4x2 + 5− x3− 3x + 4
(In short, every sign in the 2nd bracket changes.)
If you need to, shift the signed terms around so that like terms are next to eachother.
3x3 − x3 − 4x2 − 3x + 5 + 4
We now add like terms together.
2x3 − 4x2 − 3x + 9
[email protected] Factorising 39 / 43
Adding and Subtracting Polynomials
Simplify the following:
(3x3 − 4x2 + 5)− (x3 + 3x − 4)
There is a negative sign to the left of the second set of brackets. This iseffectively a multiplier of −1. From the previous workshop, we know that this −1
gets distributed to each term in the bracket.
3x3 − 4x2 + 5− (x3 + 3x − 4)
= 3x3 − 4x2 + 5− x3− 3x + 4
(In short, every sign in the 2nd bracket changes.)
If you need to, shift the signed terms around so that like terms are next to eachother.
3x3 − x3 − 4x2 − 3x + 5 + 4
We now add like terms together.
2x3 − 4x2 − 3x + 9
[email protected] Factorising 39 / 43
Adding and Subtracting Polynomials
Simplify the following:
(3x3 − 4x2 + 5)− (x3 + 3x − 4)
There is a negative sign to the left of the second set of brackets. This iseffectively a multiplier of −1. From the previous workshop, we know that this −1
gets distributed to each term in the bracket.
3x3 − 4x2 + 5− (x3 + 3x − 4)
= 3x3 − 4x2 + 5− x3− 3x + 4
(In short, every sign in the 2nd bracket changes.)
If you need to, shift the signed terms around so that like terms are next to eachother.
3x3 − x3 − 4x2 − 3x + 5 + 4
We now add like terms together.
2x3 − 4x2 − 3x + 9
[email protected] Factorising 39 / 43
Adding and Subtracting Polynomials
Simplify the following:
(3x3 − 4x2 + 5)− (x3 + 3x − 4)
There is a negative sign to the left of the second set of brackets. This iseffectively a multiplier of −1. From the previous workshop, we know that this −1
gets distributed to each term in the bracket.
3x3 − 4x2 + 5− (x3 + 3x − 4)
= 3x3 − 4x2 + 5− x3− 3x + 4
(In short, every sign in the 2nd bracket changes.)
If you need to, shift the signed terms around so that like terms are next to eachother.
3x3 − x3 − 4x2 − 3x + 5 + 4
We now add like terms together.
2x3 − 4x2 − 3x + 9
[email protected] Factorising 39 / 43
Adding and Subtracting Polynomials
Simplify the following:
−2(6x4 − 7x2 − 3) + 5(−8x6 + 3x4 − 4)
Once again, the number out the front of each brackets gets distributed to each ofthe terms.
−2(6x4 − 7x2 − 3) + 5(−8x6 + 3x4 − 4)
= −12x4 + 14x2 + 6− 40x6 + 15x4− 20
If you need to, shift the signed terms around so that like terms are next to eachother.
−12x4 + 15x4 + 14x2 + 6− 20− 40x6
We now add like terms together and write with decreasing powers:
−40x6 + 3x4 + 14x2 − 14
[email protected] Factorising 40 / 43
Adding and Subtracting Polynomials
Simplify the following:
−2(6x4 − 7x2 − 3) + 5(−8x6 + 3x4 − 4)
Once again, the number out the front of each brackets gets distributed to each ofthe terms.
−2(6x4 − 7x2 − 3) + 5(−8x6 + 3x4 − 4)
= −12x4 + 14x2 + 6− 40x6 + 15x4− 20
If you need to, shift the signed terms around so that like terms are next to eachother.
−12x4 + 15x4 + 14x2 + 6− 20− 40x6
We now add like terms together and write with decreasing powers:
−40x6 + 3x4 + 14x2 − 14
[email protected] Factorising 40 / 43
Adding and Subtracting Polynomials
Simplify the following:
−2(6x4 − 7x2 − 3) + 5(−8x6 + 3x4 − 4)
Once again, the number out the front of each brackets gets distributed to each ofthe terms.
−2(6x4 − 7x2 − 3) + 5(−8x6 + 3x4 − 4)
= −12x4
+ 14x2 + 6− 40x6 + 15x4− 20
If you need to, shift the signed terms around so that like terms are next to eachother.
−12x4 + 15x4 + 14x2 + 6− 20− 40x6
We now add like terms together and write with decreasing powers:
−40x6 + 3x4 + 14x2 − 14
[email protected] Factorising 40 / 43
Adding and Subtracting Polynomials
Simplify the following:
−2(6x4 − 7x2 − 3) + 5(−8x6 + 3x4 − 4)
Once again, the number out the front of each brackets gets distributed to each ofthe terms.
−2(6x4 − 7x2 − 3) + 5(−8x6 + 3x4 − 4)
= −12x4 + 14x2
+ 6− 40x6 + 15x4− 20
If you need to, shift the signed terms around so that like terms are next to eachother.
−12x4 + 15x4 + 14x2 + 6− 20− 40x6
We now add like terms together and write with decreasing powers:
−40x6 + 3x4 + 14x2 − 14
[email protected] Factorising 40 / 43
Adding and Subtracting Polynomials
Simplify the following:
−2(6x4 − 7x2 − 3) + 5(−8x6 + 3x4 − 4)
Once again, the number out the front of each brackets gets distributed to each ofthe terms.
−2(6x4 − 7x2 − 3) + 5(−8x6 + 3x4 − 4)
= −12x4 + 14x2 + 6
− 40x6 + 15x4− 20
If you need to, shift the signed terms around so that like terms are next to eachother.
−12x4 + 15x4 + 14x2 + 6− 20− 40x6
We now add like terms together and write with decreasing powers:
−40x6 + 3x4 + 14x2 − 14
[email protected] Factorising 40 / 43
Adding and Subtracting Polynomials
Simplify the following:
−2(6x4 − 7x2 − 3) + 5(−8x6 + 3x4 − 4)
Once again, the number out the front of each brackets gets distributed to each ofthe terms.
−2(6x4 − 7x2 − 3) + 5(−8x6 + 3x4 − 4)
= −12x4 + 14x2 + 6− 40x6
+ 15x4− 20
If you need to, shift the signed terms around so that like terms are next to eachother.
−12x4 + 15x4 + 14x2 + 6− 20− 40x6
We now add like terms together and write with decreasing powers:
−40x6 + 3x4 + 14x2 − 14
[email protected] Factorising 40 / 43
Adding and Subtracting Polynomials
Simplify the following:
−2(6x4 − 7x2 − 3) + 5(−8x6 + 3x4 − 4)
Once again, the number out the front of each brackets gets distributed to each ofthe terms.
−2(6x4 − 7x2 − 3) + 5(−8x6 + 3x4 − 4)
= −12x4 + 14x2 + 6− 40x6 + 15x4
− 20
If you need to, shift the signed terms around so that like terms are next to eachother.
−12x4 + 15x4 + 14x2 + 6− 20− 40x6
We now add like terms together and write with decreasing powers:
−40x6 + 3x4 + 14x2 − 14
[email protected] Factorising 40 / 43
Adding and Subtracting Polynomials
Simplify the following:
−2(6x4 − 7x2 − 3) + 5(−8x6 + 3x4 − 4)
Once again, the number out the front of each brackets gets distributed to each ofthe terms.
−2(6x4 − 7x2 − 3) + 5(−8x6 + 3x4 − 4)
= −12x4 + 14x2 + 6− 40x6 + 15x4− 20
If you need to, shift the signed terms around so that like terms are next to eachother.
−12x4 + 15x4 + 14x2 + 6− 20− 40x6
We now add like terms together and write with decreasing powers:
−40x6 + 3x4 + 14x2 − 14
[email protected] Factorising 40 / 43
Adding and Subtracting Polynomials
Simplify the following:
−2(6x4 − 7x2 − 3) + 5(−8x6 + 3x4 − 4)
Once again, the number out the front of each brackets gets distributed to each ofthe terms.
−2(6x4 − 7x2 − 3) + 5(−8x6 + 3x4 − 4)
= −12x4 + 14x2 + 6− 40x6 + 15x4− 20
If you need to, shift the signed terms around so that like terms are next to eachother.
−12x4 + 15x4 + 14x2 + 6− 20− 40x6
We now add like terms together and write with decreasing powers:
−40x6 + 3x4 + 14x2 − 14
[email protected] Factorising 40 / 43
Adding and Subtracting Polynomials
Simplify the following:
−2(6x4 − 7x2 − 3) + 5(−8x6 + 3x4 − 4)
Once again, the number out the front of each brackets gets distributed to each ofthe terms.
−2(6x4 − 7x2 − 3) + 5(−8x6 + 3x4 − 4)
= −12x4 + 14x2 + 6− 40x6 + 15x4− 20
If you need to, shift the signed terms around so that like terms are next to eachother.
−12x4 + 15x4 + 14x2 + 6− 20− 40x6
We now add like terms together and write with decreasing powers:
−40x6 + 3x4 + 14x2 − 14
[email protected] Factorising 40 / 43
Multiplying Polynomials
We can use the distributive law to multiply two polynomials together
(3x2 + 2x)(4x5 + 3x) =
12x7 + 9x3+ 8x6+ 6x2
Every term in the first bracket meets every term in the second bracket.(We saw this in the previous workshop.)
Sometimes, we work with “longer” polynomials. The rule is still the same. Everyterm in the first bracket must meet every term in the second bracket.
[email protected] Factorising 41 / 43
Multiplying Polynomials
We can use the distributive law to multiply two polynomials together
(3x2 + 2x)(4x5 + 3x) =
12x7 + 9x3+ 8x6+ 6x2
Every term in the first bracket meets every term in the second bracket.(We saw this in the previous workshop.)
Sometimes, we work with “longer” polynomials. The rule is still the same. Everyterm in the first bracket must meet every term in the second bracket.
[email protected] Factorising 41 / 43
Multiplying Polynomials
We can use the distributive law to multiply two polynomials together
(3x2 + 2x)(4x5 + 3x) = 12x7
+ 9x3+ 8x6+ 6x2
Every term in the first bracket meets every term in the second bracket.(We saw this in the previous workshop.)
Sometimes, we work with “longer” polynomials. The rule is still the same. Everyterm in the first bracket must meet every term in the second bracket.
[email protected] Factorising 41 / 43
Multiplying Polynomials
We can use the distributive law to multiply two polynomials together
(3x2 + 2x)(4x5 + 3x) = 12x7
+ 9x3+ 8x6+ 6x2
Every term in the first bracket meets every term in the second bracket.(We saw this in the previous workshop.)
Sometimes, we work with “longer” polynomials. The rule is still the same. Everyterm in the first bracket must meet every term in the second bracket.
[email protected] Factorising 41 / 43
Multiplying Polynomials
We can use the distributive law to multiply two polynomials together
(3x2 + 2x)(4x5 + 3x) = 12x7 + 9x3
+ 8x6+ 6x2
Every term in the first bracket meets every term in the second bracket.(We saw this in the previous workshop.)
Sometimes, we work with “longer” polynomials. The rule is still the same. Everyterm in the first bracket must meet every term in the second bracket.
[email protected] Factorising 41 / 43
Multiplying Polynomials
We can use the distributive law to multiply two polynomials together
(3x2 + 2x)(4x5 + 3x) = 12x7 + 9x3
+ 8x6+ 6x2
Every term in the first bracket meets every term in the second bracket.(We saw this in the previous workshop.)
Sometimes, we work with “longer” polynomials. The rule is still the same. Everyterm in the first bracket must meet every term in the second bracket.
[email protected] Factorising 41 / 43
Multiplying Polynomials
We can use the distributive law to multiply two polynomials together
(3x2 + 2x)(4x5 + 3x) = 12x7 + 9x3+ 8x6
+ 6x2
Every term in the first bracket meets every term in the second bracket.(We saw this in the previous workshop.)
Sometimes, we work with “longer” polynomials. The rule is still the same. Everyterm in the first bracket must meet every term in the second bracket.
[email protected] Factorising 41 / 43
Multiplying Polynomials
We can use the distributive law to multiply two polynomials together
(3x2 + 2x)(4x5 + 3x) = 12x7 + 9x3+ 8x6
+ 6x2
Every term in the first bracket meets every term in the second bracket.(We saw this in the previous workshop.)
Sometimes, we work with “longer” polynomials. The rule is still the same. Everyterm in the first bracket must meet every term in the second bracket.
[email protected] Factorising 41 / 43
Multiplying Polynomials
We can use the distributive law to multiply two polynomials together
(3x2 + 2x)(4x5 + 3x) = 12x7 + 9x3+ 8x6+ 6x2
Every term in the first bracket meets every term in the second bracket.(We saw this in the previous workshop.)
Sometimes, we work with “longer” polynomials. The rule is still the same. Everyterm in the first bracket must meet every term in the second bracket.
[email protected] Factorising 41 / 43
Multiplying Polynomials
We can use the distributive law to multiply two polynomials together
(3x2 + 2x)(4x5 + 3x) = 12x7 + 9x3+ 8x6+ 6x2
Every term in the first bracket meets every term in the second bracket.(We saw this in the previous workshop.)
Sometimes, we work with “longer” polynomials. The rule is still the same. Everyterm in the first bracket must meet every term in the second bracket.
[email protected] Factorising 41 / 43
Multiplying Polynomials
We can use the distributive law to multiply two polynomials together
(3x2 + 2x)(4x5 + 3x) = 12x7 + 9x3+ 8x6+ 6x2
Every term in the first bracket meets every term in the second bracket.(We saw this in the previous workshop.)
Sometimes, we work with “longer” polynomials. The rule is still the same. Everyterm in the first bracket must meet every term in the second bracket.
[email protected] Factorising 41 / 43
Expansion
In general
(polynomial) × (polynomial)
expands out to
polynomial
Note: We saw earlier that 2nd order polynomials can (sometimes) be factorised.In general, polynomials can also (sometimes) be be factorised.
However, that’s a topic for another time . . .
[email protected] Factorising 42 / 43
Expansion
In general
(polynomial) × (polynomial)
expands out to
polynomial
Note: We saw earlier that 2nd order polynomials can (sometimes) be factorised.In general, polynomials can also (sometimes) be be factorised.
However, that’s a topic for another time . . .
[email protected] Factorising 42 / 43
Expansion
In general
(polynomial) × (polynomial)
expands out to
polynomial
Note: We saw earlier that 2nd order polynomials can (sometimes) be factorised.
In general, polynomials can also (sometimes) be be factorised.
However, that’s a topic for another time . . .
[email protected] Factorising 42 / 43
Expansion
In general
(polynomial) × (polynomial)
expands out to
polynomial
Note: We saw earlier that 2nd order polynomials can (sometimes) be factorised.In general, polynomials can also (sometimes) be be factorised.
However, that’s a topic for another time . . .
[email protected] Factorising 42 / 43
Expansion
In general
(polynomial) × (polynomial)
expands out to
polynomial
Note: We saw earlier that 2nd order polynomials can (sometimes) be factorised.In general, polynomials can also (sometimes) be be factorised.
However, that’s a topic for another time . . .
[email protected] Factorising 42 / 43
Using STUDYSmarter Resources
This resource was developed for UWA students by the STUDYSmarter team forthe numeracy program. When using our resources, please retain them in their
original form with both the STUDYSmarter heading and the UWA crest.
[email protected] Factorising 43 / 43