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: ________________________