unit 2 lesson 1: greatest common factor (gcf) · unit 2 lesson 1: greatest common factor (gcf)...
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Unit 2 – Lesson 1: Greatest Common Factor (GCF)
Factoring polynomials with a greatest
common factor (GCF)
What is a GCF?
GCF of _____________________________ is the
biggest whole number that is a factor
of ALL coefficients in given problem.
GCF of ______________________________is the
smallest power of the variable from
each term. *If variable is missing from
any term, it is NOT a part of the GCF.
***** Always look for the _______________
before using any other factoring
method.
Identify the GCF between the given
terms:
1)24𝑥2 𝑎𝑛𝑑 16𝑥
GCF: ________________________________________
2)30𝑥2𝑦 𝑎𝑛𝑑 36𝑥3
GCF: ________________________________________
3)30𝑥2𝑦3𝑧, 12𝑥𝑦𝑧2 𝑎𝑛𝑑 6𝑥4𝑦2𝑧
GCF: ________________________________________
Steps for Factoring using GCF:
1. Find the GCF of all terms
2. Divide each term by the GCF
3. Put GCF in front of terms in
parenthesis
Factor by finding GCF
1. 8𝑥3 − 8
Factored Form:_____________________________
2. 3𝑥3 − 6𝑥2 − 24𝑥
Factored Form:_____________________________
3. 21𝑝6 + 30𝑝2 + 27
Factored Form:_____________________________
4. 30𝑚6 + 15𝑚𝑛2 − 25
Factored Form:_____________________________
5. 30𝑦4𝑧3𝑥5 + 50𝑦4𝑧5 − 10𝑦4𝑧3𝑥
Factored Form:_____________________________
6. 5𝑥2 − 10𝑥 + 35
Factored Form:______________________________
7. 6𝑐3𝑑 − 12𝑐2𝑑2 + 3𝑐𝑑
Factored Form:______________________________
8. 6𝑥3 + 3𝑥2 − 12𝑥
Factored Form:______________________________
9. 16𝑥3𝑦4𝑧 − 8𝑥2𝑦2𝑧3 + 12𝑥𝑦3𝑧2
Factored Form:______________________________
Unit 2 – Lesson 2: Factoring Quadratics (a=1)
Factoring Quadratic Equations
a=1
* To ___________________________ a trinomial
means to write it as the product of two
binomials
*Check for ______________ FIRST!
1. 𝑥2 + 6𝑥 + 8
a=___________ b=___________ c=___________
GCF: ___________________________
Factored Form:_____________________________
2. 𝑥2 − 3𝑥 + 2
GCF: ___________________________
Factored Form:_____________________________
3. 𝑥2 − 2𝑥 − 8
GCF: ___________________________
Factored Form:_____________________________
4. 𝑥2 − 5𝑥 − 14
GCF: ___________________________
Factored Form:_____________________________
5. 𝑥2 − 16𝑥 + 64
GCF: ___________________________
Factored Form:_____________________________
6. 𝑥2 − 𝑥 − 42
GCF: ___________________________
Factored Form:_____________________________
7. 𝑥2 − 5𝑥 − 6
GCF: ___________________________
Factored Form:_____________________________
8. 𝑥2 + 6𝑥 − 7
GCF: ___________________________
Factored Form:_____________________________
9. 2𝑥2 − 6𝑥 − 140
GCF: ___________________________
Factored Form:_____________________________
10. 3𝑥2 − 21𝑥 + 18
GCF: ___________________________
Factored Form:_____________________________
Lesson 2 Practice
1. 𝑚2 − 3𝑚 − 18
GCF: ___________________________
Factored Form:_____________________________
2. 𝑘2 − 14𝑘 + 45
GCF: ___________________________
Factored Form:_____________________________
3. 𝑥2 − 11𝑥 + 30
GCF: ___________________________
Factored Form:_____________________________
4. 𝑝2 + 14𝑝 + 45
GCF: ___________________________
Factored Form:_____________________________
5. 𝑥2 − 3𝑥 + 2 GCF: ___________________________
Factored Form:_____________________________
6. 4𝑣2 + 44𝑣 + 120
GCF: ___________________________
Factored Form:_____________________________
7. 3𝑝2 + 12𝑝 − 54
GCF: ___________________________
Factored Form:_____________________________
8. 𝑚2 + 14𝑚 + 40
GCF: ___________________________
Factored Form:_____________________________
9. 2𝑥2 + 12𝑥 − 54
GCF: ___________________________
Factored Form:_____________________________
Unit 2 – Lesson 3: Factoring Quadratics (a≠1)
Factoring Quadratic Equations
a≠1
*Solve the same way as when
a=1, but then divide
____________________ by ______________.
This method is called
_____________________________________.
1. 6𝑥2 + 13𝑥 − 5
a=___________ b=___________ c=___________
GCF: _______________________
Does a=1? __________________
Factored Form:_____________________________
2. 6𝑥2 + 16𝑥 − 6
GCF: _______________________
Does a=1? __________________
Factored Form:_____________________________
3. 3𝑥2 + 29𝑥 + 40
GCF: _______________________
Does a=1? __________________
Factored Form:_____________________________
4. 7𝑥2 + 44𝑥 − 35
GCF: _______________________
Does a=1? __________________
Factored Form:_____________________________
5. 4𝑥2 + 11𝑥 + 6
GCF: _______________________
Does a=1? __________________
Factored Form:_____________________________
6. 3𝑥2 + 11𝑥 − 20
GCF: _______________________
Does a=1? __________________
Factored Form:_____________________________
Lesson 3 Practice
7. −3𝑥2 + 16𝑥 + 12
GCF: _______________________
Does a=1? __________________
Factored Form:_____________________________
8. 3𝑥2 − 2𝑥 − 5
GCF: _______________________
Does a=1? __________________
Factored Form:____________________________
9. 9𝑥2 + 66𝑥 + 21
GCF: _______________________
Does a=1? __________________
Factored Form:_____________________________
10. 4𝑥2 − 35𝑥 + 49
GCF: _______________________
Does a=1? __________________
Factored Form:_____________________________
11. 16𝑥2 + 60𝑥 − 100
GCF: _______________________
Does a=1? __________________
Factored Form:_____________________________
12. 2𝑥2 − 15𝑥 + 7
GCF: _______________________
Does a=1? __________________
Factored Form:___________________________
Unit 2 – Lesson 4: Solving by Factoring Quadratic Functions (GCF, a=1, a≠1)
Steps to Solving Quadratic Functions
by Factor
1. Get all terms on one side of the equal
sign in standard form and set equal to
zero.
2. Determine if there is a GCF and factor
out GCF if there is one. 3. Determine if a=1 or if a≠1 and factor accordingly.
4. Once your equation is fully factored, set
each term equal to zero and solve.
Zero Product Property If a and b are real numbers and if a∙=0, then a=0 or b=0
Example: x(x – 7) = 0
x=0 x – 7 = 0 x= 7
1. 𝑥2 − 3𝑥 = 18
GCF: _______________________
Does a=1? __________________
Factored Form:_____________________________
Solution: ___________________________________
2. 3𝑥2 + 7𝑥 = 6
GCF: _______________________
Does a=1? __________________
Factored Form:_____________________________
Solution: ___________________________________
3. 9𝑥2 − 24𝑥 + 16 = 0
GCF: _______________________
Does a=1? __________________
Factored Form:_____________________________
Solution: ___________________________________
4. 2𝑥3 − 18𝑥 = 0
GCF: _______________________
Does a=1? __________________
Factored Form:_____________________________
Solution: ___________________________________
5. 𝑥2 + 𝑥 = 30
GCF: _______________________
Does a=1? __________________
Factored Form:_____________________________
Solution: ___________________________________
6. 5𝑥2 − 10𝑥 − 175 = 0
GCF: _______________________
Does a=1? __________________
Factored Form:_____________________________
Solution: ___________________________________
7. 𝑥2 + 56 = 15𝑥
GCF: _______________________
Does a=1? __________________
Factored Form:_____________________________
Solution: ___________________________________
8. 2𝑥2 − 5𝑥 − 12 = 0
GCF: _______________________
Does a=1? __________________
Factored Form:_____________________________
Solution: ___________________________________
Lesson 5: Solving Quadratics by Completeing the Square If a quadratic equation has no linear term,
you can use square roots to solve it. By
definition, if x2 = c, then x = √c and x = -
√c, usually written x = √c
To solve a quadratic equation using
square roots:
o Isolate the squared term
o Take the square root of both
sides
Review:
1. 𝑥2 = 256 Solution:
2. 6𝑥2 = 150 Solution:
3. 2𝑥2 − 12 = 132 Solution:
4. (𝑥 + 3)2 = 9 Solution:
5. 3(𝑥 − 2)2 + 4 = 52 Solution:
Solving quadratic equations by completing
the square
Ex: 𝑥2 + 6𝑥 + 9
Ex: 𝑥2 − 10𝑥 + 25
Ex: 𝑥2 + 12𝑥 + 36 To create a perfect square trinomial:
Move terms with variables to the left side of the
equal sign
Move terms constants to the right side of the equal
sign
Find the constant term by squaring half the
coefficient of the linear term
(𝑏2)2=the constant term
Ex 1: 𝑥2 + 20𝑥 + ___________ Ex 2: 𝑥2 + 5𝑥 + ____________
Solve the following equation by
completing the square
1. 𝑥2 + 8𝑥 − 20 = 0
GCF: _______________________
Does a=1? __________________
Factored Form:_____________________________
Solution: ___________________________________
2. 𝑥2 + 2𝑥 − 84 = 0
GCF: _______________________
Does a=1? __________________
Factored Form:_____________________________
Solution: ___________________________________
3. 𝑥2 + 8𝑥 − 84 = 0
GCF: _______________________
Does a=1? __________________
Factored Form:_____________________________
Solution: ___________________________________
4. 𝑥2 + 25 = 10𝑥
GCF: _______________________
Does a=1? __________________
Factored Form:_____________________________
Solution: ___________________________________
5. 2𝑥2 + 16𝑥=128
GCF: _______________________
Does a=1? __________________
Factored Form:_____________________________
Solution: ___________________________________
Lesson 5 Practice
Solve by completing the square
1. 𝑥2 + 6𝑥 = 16
GCF: _______________________
Does a=1? __________________
Factored Form:_____________________________
Solution: ___________________________________
2. 𝑥2 + 16𝑥 − 7 = 0
GCF: _______________________
Does a=1? __________________
Factored Form:_____________________________
Solution: ___________________________________
3. 𝑥2 + 2𝑥 − 6 = 0
GCF: _______________________
Does a=1? __________________
Factored Form:_____________________________
Solution: ___________________________________
4. 𝑥2 − 9𝑥 + 18 = 0
GCF: _______________________
Does a=1? __________________
Factored Form:_____________________________
Solution: ___________________________________
5. 𝑥2 + 11𝑥 = −24
GCF: _______________________
Does a=1? __________________
Factored Form:_____________________________
Solution: ___________________________________
6. 2𝑥2 + 4𝑥 = 5
GCF: _______________________
Does a=1? __________________
Factored Form:_____________________________
Solution: ___________________________________
7. 4𝑥2 − 𝑥 − 3 = 0
GCF: _______________________
Does a=1? __________________
Factored Form:_____________________________
Solution: ___________________________________
8. 3𝑥2 − 16𝑥 − 35 = 0
GCF: _______________________
Does a=1? __________________
Factored Form:_____________________________
Solution: ___________________________________
9. 3𝑥2 − 11𝑥 − 4 = 0
GCF: _______________________
Does a=1? __________________
Factored Form:_____________________________
Solution: ___________________________________
Lesson 6: Solving Quadratic Equations by using Quadratic Formula Recall: Quadratic Equation
-We use quadratic formula when we cannot
factor (when the given quadratic equation
is not factorable). Make sure the equation
equals 0 before you use the Quadratic
Formula!
____________________________________
____________________________________
1: Find the factors of 3𝑥2 − 13𝑥 + 4 = 0
2: Find the factors of 3𝑥2 + 4𝑥 + 2 = 0
3: Find the factors of 2𝑥2 − 4𝑥 − 3 = 0
4: Find the factors of 2𝑥2 + 𝑥 = −1
5: Find the factors of −6𝑥 + 2𝑥2 = −10
Solve
1. Solve for x in 𝑥2 + 81 = 0
2. Solve for x in 𝑥2 + 27 = 0
3. Solve for x in 4𝑥2 − 96 = 0
a = _______
b = _______
c = _______
a = _______
b = _______
c = _______
a = _______
b = _______
c = _______
a = _______
b = _______
c = _______
a = _______
b = _______
c = _______
Lesson 6 Practice Solve each equation using the
Quadratic Formula.
1. 𝑥2 − 4𝑥 + 3 = 0
Solution:______________________________
2. 2𝑥2 + 5𝑥 − 7 = 0
Solution:______________________________
3. 3𝑥2 + 2𝑥 − 1 = 0
Solution:______________________________
4. 𝑥2 = 6𝑥 − 1
Solution:______________________________
5. 2𝑥2 = 12𝑥 − 8
Solution:______________________________
6. 2𝑥2 − 7𝑥 − 13 = −10
Solution:______________________________
7. 4𝑥2 + 8𝑥 + 7 = 4
Solution:______________________________
8. 5𝑥2 + 9𝑥 = −4
Solution:______________________________
9. 9𝑥2 = 4 + 7𝑥
Solution:______________________________
Unit 2: Quadratics Review Name: _________________________________
U2L1: I can factor the GCF from trinomial expressions.
1) 6𝑥3𝑦2 + 12𝑥𝑦3 − 36𝑦2 2) −2𝑏3𝑝 + 4𝑏3𝑝𝑤 + 24𝑏2𝑝2
GCF: ______________ GCF: ______________
Factored Form: _______________________ Factored Form: _______________________
U2L2: I can factor a trinomial expression when a=1.
1) 𝑥2 − 3𝑥 − 40 2) 3𝑥2 + 9𝑥 − 12
GCF: ______________ GCF: ______________
Factored Form: _______________________ Factored Form: _______________________
U2L3: I can factor a trinomial expression when a≠1.
1) 7𝑥2 − 52𝑥 + 21 2) 16𝑥2 − 72𝑥 + 80
GCF: ______________ GCF: ______________
Does a=1? __________________________ Does a=1? __________________________
a= _______ b= _______ c= _______ a= _______ b= _______ c= _______
Factored Form: _______________________ Factored Form: _______________________
U2L4: I can solve quadratic equations using factoring methods: GCF, a=1, a≠1.
1) 𝑥2 − 4𝑥 + 3 = 0 2) 4𝑥2 + 32𝑥 = −64
GCF: ______________ GCF: ______________
Does a=1? __________________________ Does a=1? __________________________
a= _______ b= _______ c= _______ a= _______ b= _______ c= _______
Factored Form: _______________________ Factored Form: _______________________
Solutions: ____________________________ Solutions: ____________________________
3) 2𝑥2 − 5𝑥 + 3 = 0 4) 4𝑥2 − 6𝑥 + 2 = 0
GCF: ______________ GCF: ______________
Does a=1? __________________________ Does a=1? __________________________
a= _______ b= _______ c= _______ a= _______ b= _______ c= _______
Factored Form: _______________________ Factored Form: _______________________
Solutions: ____________________________ Solutions: ____________________________
U2L5: I can solve quadratic equations using the quadratic formula.
1) 𝑥2 + 4𝑥 − 60 = 0 2) 6𝑥2 + 2𝑥 = −5
a = _________ a = _________
b = _________ b = _________
c = _________ c = _________
Solutions: ________________________ Solutions: ________________________
U2L6: I can solve quadratic equations by completing the square.
1) 𝑥2 − 4𝑥 − 21 = 0 2)𝑥2 + 14𝑥 + 40 = 0
Solutions: ________________________ Solutions: ________________________