Creating brackets

In this powerpoint, we meet 5 different methods of factorising.

Type 1 – Common Factor

Type 2 – Difference of Two Squares

Type 3 – Grouping

This involves taking a term outside the brackets. Always try to do this first.Try this when you have two terms with a minus between

This is the easiest one to pick – use it when there are 4 terms!

Types 4 and 5

Quadratic trinomials

Use these for expressions with 3 terms.

They will be of the format

x2 + bx + c (Type 4) OR

ax2 + bx + c (Type 5)

Where a, b and c are just numbers

Factorising just makes me sooooo happy!!

Summary

TypeType When to UseWhen to Use

1. Common 1. Common factorfactor

Always try first before any other methodAlways try first before any other method Examples: Examples: aa22 – 9 – 9aa ; 2 ; 2xy xy + 5+ 5xx22

2. Difference 2. Difference of Two of Two squaressquares

When there are only When there are only 2 terms2 terms which are which are squaressquares There must be a There must be a minus signminus sign Examples: Examples: aa22 – 25 ; 81 – 4 – 25 ; 81 – 4bb22 ; ; ww44 – 16 – 16

3. Grouping3. Grouping There are There are 4 terms.4 terms. Example: Example: aa22 – 4 – 4aa + 3 + 3ab – ab – 1212bb

4. Quadratic 4. Quadratic Trinomial (I)Trinomial (I)

There are There are 3 terms. Has a squared term.3 terms. Has a squared term. Examples: Examples: aa22 – 9 – 9aa + 20 ; 6 – 5 + 20 ; 6 – 5bb + + bb22

5. Quadratic 5. Quadratic Trinomial (II)Trinomial (II)

There are There are 3 terms. Has a squared term 3 terms. Has a squared term with a number attached in front.with a number attached in front. Examples: 2Examples: 2aa22 – 3 – 3aa – 5 ; 6 – 5 ; 6bb – 5 – 5bb22 + 3 + 3bb

Type 1 of 5 – common factor

Always try this first, regardless of what type it is

3a – 12 = 3(a – 4)

3a2 – 12a =

3a2 + 6a + 12 =

20ab – 12b2 =

30a6 – 15a5 =

3a(a – 4)

4b(5a – 3b)

15a5(2a – 1)

3(a2 + 2a + 4)

Remember – take out the largest factor you can!

Always look for a

common factor!

Type 2 of 5 – diff of 2 squares

To qualify as a Type 2, an expression

• must have only 2 terms which are SQUARES

• must have a MINUS sign separating them

Examples

a2 – 9 = (a – 3)(a + 3)

16 – a2 = (4 – a)(4 + a)

(2b)2 – (3a)2 =

9b2 – 25 = (3b – 5)(3b + 5)

(2b – 3a)(2b + 3a)

Combining Types 1 and 2

Example 1 .....Factorise 5x2 – 45

STEP 1 Treat as a Type 1, and take out common factor first, 5Write 5(x2 – 9)

STEP 2 Now do expression in brackets as a Type 2

Write 5(x – 3)(x + 3)...ANS!

LookMum ! It’s a

difference of 2

squares!

Example 2 .....Factorise x4 – 81

STEP 1 Treat as a Type 2, and write as difference of 2 squares.....(x2 – 9)(x2 + 9)

STEP 2

(x2 – 9)(x2 + 9)

(x – 3)(x + 3)(x2 + 9)....ANS!!

Now check out the thing in each bracket. We can factorise the first one, but not the second.

Y’can’t factorise a SUM of two squares Stupid! x2 + 9 has to stay as it is. It’s not

the same as (x + 3)(x + 3) is it now???

Example 3 .....Factorise 80a4 – 405b12

STEP 2

STEP 3

STEP 1 Identify common factor, 5 and remove

Write 5(16a4 – 81b12)

Now work on the terms in the brackets

This is a difference of 2 squares and becomes (4a2 – 9b6) (4a2 + 9b6)

Now work on the terms in the 1st bracket.

This is a difference of 2 squares and becomes (2a – 3b3) (2a + 3b3) . Write

Write 5(4a2 – 9b6) (4a2 + 9b6)

5(2a – 3b3) (2a + 3b3) (4a2 + 9b6)

Example 4 .....Factorise 9a2 – (x – 2a)2

Just treat as difference of 2 squares of the format

9a2 – b2 where the b = [x – 2a]

Factorising it then becomes

= (3a – b)(3a + b)And then replacing the b with [x – 2a] we get

= (3a – [x – 2a])(3a + [x – 2a])Now get rid of square brackets

= (3a – x + 2a)(3a + x – 2a)Clean up

= (5a – x )(a + x) Ans!!You could check your answer by expanding it and also expanding the original question. They should both give the same thing.

Type 3 of 5 – Grouping You can tell when you’ve got one of these because

there are FOUR TERMS !!!

Example 1Factorise 2a – 4b + ax – 2bx

STEP 1 – split it into “2 by 2” = 2a – 4b + ax – 2bx

STEP 2 – factorise each pair separately as Type 1 = 2(a – 2b) + x(a – 2b)

STEP 3 – take out the (a – 2b) as a common factor

= (a – 2b)(2 + x)...ans!!

No need to be confused!

Type 3 of 5 – Grouping

Example 2

Factorise xy + 5x – 2y – 10

STEP 1 – split it into “2 by 2” = xy + 5x – 2y – 10

STEP 2 – factorise each pair separately as Type 1 = x(y + 5) – 2 (y + 5)

STEP 3 – take out the (y + 5) as a factor

= (y + 5)(x – 2) ans!!

If these are the same, it’s a good

sign!

Type 3 of 5 – Grouping

Example 3

Factorise x2 – x – 5x + 5

STEP 1 – split it into “2 by 2” = x2 – x – 5x + 5

STEP 2 – factorise each pair separately as Type 1 = x(x – 1) – 5 (x – 1)

STEP 3 – take out the (x – 1) as a factor

= (x – 1 )(x – 5) ans!!

Ewbewdy!!They’re the same! On my way to a VHA

Example 4 - harder

Factorise x2 – 4y2 – 2ax – 4ay

STEP 1 – split it into “2 by 2”

= x2 – 4y2 – 2ax – 4ay

STEP 2 – factorise each pair separately

= (x – 2y) (x + 2y)

STEP 3 – take out the (x + 2y) as a factor

= (x + 2y)(x – 2y – 2a) ans!!

– 2a (x + 2y)

1st pair – Type 2

2nd pair – Type 1

Awwright! They’re the

same!!

Type 4 of 5 – Easy Quadratic Trinomial

Example 1 .....Factorise x2 + 5x + 6

You can usually pick these as they have 3 TERMS

STEP 1 – Make 2 brackets

(x..............)(x.............)

STEP 2 – Look for 2 numbers that

Multiply to make +6

Add to make +5 +2 & +3

STEP 3 – Put ‘em in the brackets (x + 2)(x + 3)

Type 4 of 5 – Easy Quadratic Trinomial

Example 2 .....Factorise 2x2 – 6x – 20 STEP 1 – take out a common factor (remember this should be your 1st step EVERY time!!)

= 2(x2 – 3x – 10)

STEP 2 – Ignore the 2. For the expression inside the brackets, look for 2 numbers that

Multiply to make – 10

Add to make – 3 +2 & – 5

STEP 3 – Put ‘em in the brackets2(x + 2)(x – 5)

Type 4 of 5 – Easy Quadratic Trinomial

Example 3 .....Factorise 6 + 5x – x2 STEP 1 – Rearrange into “normal” format with x2 at the front, then x, then the number

= – x2 + 5x + 6

STEP 2 – Now take out a common factor – 1

STEP 3 – Ignore the minus. Look for 2 numbers that add to – 5, and multiply to – 6.

= – (x2 – 5x – 6)

These are +1 and –6. – (x + 1)(x – 6)

Type 5 of 5 – Harder Quadratic Trinomial

Example 1 .....Factorise 2x2 + 5x – 3 STEP 1 – Draw up a fraction like this

2........)2........)(2( xx

STEP 2 – Look for two numbers that

ADD to make +5

MULT to make – 6

2 × – 3 = – 6

Numbers are +6, – 1 2

)12)(62(

xx

= (x + 3)(2x – 1) ANSNote the 2 in bottom must cancel one whole bracket FULLY! So (2x + 6) becomes (x + 3)

With a number in front of the x2

Type 5 of 5 – Harder Quadratic Trinomial

Example 2 .....Factorise 3x2 + 8x – 3 STEP 1 – Draw up a fraction like this

3........)3........)(3( xx

STEP 2 – Look for two numbers that

ADD to make +8

MULT to make – 9

3 × – 3 = – 9

Numbers are +9, – 1 3

)13)(93(

xx

= (x + 3)(3x – 1) ANSNote the 3 in bottom must cancel one whole bracket FULLY! So (3x + 9) becomes (x + 3)

With a number in front of the x2

Type 5 of 5 – Harder Quadratic Trinomial

Example 3 .....Factorise 6x2 – 19x + 10 STEP 1 – Draw up a fraction like this

6........)6........)(6( xx

STEP 2 – Look for two numbers that

ADD to make –19

MULT to make 60

6 × 10 = 60

Numbers are –4 , –15 32

)156)(46(

xx

= (3x – 2)(2x – 5) ANSNote the 6 in bottom would not cancel either bracket FULLY! So we broke the 6 into 2 x 3 then cancelled.

With a number in front of the x2

Now wozn’t that just a barrel of fun??