W8W-REVIEW FOR FINAL--Factoring Polynomials

Factoring Polynomials

Multiple Choice

Which of the following represents the factors of 12?

• 1, 12

• 1, 2, 3, 4, 6, 12

• 12, 24, 36, 48, 60

• No Factors

Multiple Choice

Which of the following represents the factors of 36?

• 1, 2, 3, 4, 6, 9, 12, 18, 36

• 36, 52, 108, 144

• No Factors

Open Ended

What does it mean when you have something in common? How can we apply that to math?

Greatest Common Factor

Definition : The biggest number that is a factor of each number!

Fill in the Blank

What is the Greatest Common Factor of 10 and 5?

Poll

Which of the following do you think is the GCF of

 $12x^3+3x^2$  

•  $3x$  

• 3

•  $3x^2$  

Multiple Choice

What is the GCF of this expression:

 $10x^2-5$  

• 10

• x

• 5

Multiple Choice

What is the GCF of this expression:

 $14r^{22}+7r^{22}$  

•  $7r^{22}$  

• 15

• No GCF

Multiple Choice

Which of the following is equivalent to w³ + 3w

• w (w²+3)

• 3w (w²+1)

• There is no GCF

Multiple Choice

Which of the following is equivalent to

$$4r^5+2r^3$$  

• $$r^3\left(4r^2+2\right)$$  

• $$2r^3\left(2r^2+1\right)$$  

• There is no GCF

Factoring: Expressions with 4 Terms Lesson

By Grouping and Box Method

Some foundational information

Vocabulary

Binomial Factor

Binomial factors are polynomial factors that have exactly two terms. (x-r) represents a binomial factor.

Factored Form

Factored form, the product of a constant and two linear terms:...a is the constant... (x-r) is a binomial factor...r is the root/x-intercept. Converting a quadratic function to factored form is called factoring.

Factor By Grouping Video Link

https://www.youtube.com/watch?v=NYMpf4fdnbY

Factor by Grouping Example

• Divide the expression into two groups inside ( ).

• Each part should have a common binomial factor...same two terms inside ()

• Factor out the Common Binomials

• You should have the factored form of the expression...two binomial factors

Multiple Choice

Factor by grouping

x³ - 4x² - 4x + 16

Hint: the answer is not completely factored

• (x + 4)(x² - 4)

• (x + 4)(x + 2)

• (x - 4)(x - 2)

• (x - 4)(x² - 4)

Multiple Choice

Factor by grouping:

4p³ + 8p² + 3p + 6

• (2p² + 6)(p + 4)

• (2p² + 3)(p + 2)

• (4p² + 8p)(3p + 6)

• (4p² + 3)(p + 2)

Multiple Choice

Factor

$$16mn-4m^2+28n-7m$$  by grouping

• (4m+7)(4n-m)

• (4m-7)(4n-m)

• (4m-7)(4n+m)

The Box Method

When given a Quadratic Trinomial

Box Method Vedeo Link

https://www.youtube.com/watch?v=OKkg2kUsDq4&t=44s

Box Method

• Factor the GCF from each row and column

• The column and row of factors will be the binomial factors of the trinomial.

• (x+2)(2x+5) is the factored form of the original expression

Multiple Choice

What expressions should replace A and B?

• A = 2 and B = -1

• A = 2x and B = -1

• A = 2x² and B= -5

• A = 2x and B = -5

Multiple Choice

Factor using box method

x³ - 4x² - 4x + 16

Hint: the answer is not completely factored

• (x + 4)(x² - 4)

• (x + 4)(x + 2)

• (x - 4)(x - 2)

• (x - 4)(x² - 4)

Multiple Choice

Factor by box method:

4p³ + 8p² + 3p + 6

• (2p² + 6)(p + 4)

• (2p² + 3)(p + 2)

• (4p² + 8p)(3p + 6)

• (4p² + 3)(p + 2)

Multiple Choice

Factor

$$16mn-4m^2+28n-7m$$  by grouping

• (4m+7)(4n-m)

• (4m-7)(4n-m)

• (4m-7)(4n+m)

Multiple Choice

Given the expression

$$3x^2+13x+12$$ . Determine the values that would fill in boxes B and C.

• 8 and 5

• 9 and 4

• 15 and -2

• 3 and 12

Multiple Choice

3x² + 8x + 5

• (x - 1) (3x - 5)

• (3x - 1) (x - 5)

• (x + 1) (3x + 5)

• (3x + 1) (x + 5)

Multiple Choice

Factor: 2x² - 5x + 2

• (2x + 1)(x - 2)

• (2x - 2)(x - 1)

• (2x - 1)(x - 2)

• (2x + 2)(x - 1)

Multiple Choice

Factor 12xy − 28x − 15y + 35

• (4x-5)(3y+7)

• (4x+5)(3y-7)

• (4x-5)(3y-7)

Difference of Two Squares & Perfect Square Trinomials

Perfect Square Trinomials

What is a Perfect Square Trinomial?

Factor x² + 8x + 16

• Write the expression x² + 8x + 16 as a² + 2ab + b²

• x² + 8x + 16 ⟹ (x)² + 2 (x) (4) + (4)²

• = (x + 4)²

• = (x + 4) (x + 4)

• 
  

Factor x²+ 6x + 9

Solution

We can rewrite the expression x² + 6x + 9 in the form a² + 2ab + b² as;

x²+ 6x + 9 ⟹ (x)² + 2 (x) (3) + (3)²

Applying the formula of a² + 2ab + b² = (a + b)² to the expression gives;

= (x + 3)²

= (x + 3) (x + 3)

Factor 4a² – 4ab + b²

• 4a² – 4ab + b² ⟹ (2a)² – (2)(2) ab + b²

  
  

  
  

• = (2a – b)²

• = (2a – b) (2a – b)

• 
  

Multiple Choice

Which of the following are the factors of x² + 10x + 25?

• (x+5) (x+5)

• (x+5) (x-5)

• (x-5) (x-5)

• (x+10) (x+15)

Multiple Choice

What are the factors of m² - 6m + 9?

• (m-3) (m-6)

• (m+3) (m+3)

• (m+3) (m-3)

• (m-3) (m-3)

Multiple Choice

Find the factors of 4x² + 20x + 25

• (4x+5) (4x+5)

• (4x-5) (4x-5)

• (2x+5) (2x+5)

• (2x-5) (2x-5)

Multiple Select

Is x² - 20x + 100 a perfect square trinomial?

• Yes

• No

Multiple Select

Is 2x²+ 8x + 8 factorable?

• Yes

• No

Difference of Two Squares

Multiple Choice

Factor the following expression:

$$x^2-81$$  

• (x-9)(x-9)

• (x+9)(x+9)

• (x+9)(x-9)

• Not Factorable

Multiple Choice

Factor the following expression:

$$x^2-144$$  

• (x-12)(x+12)

• (x-12)(x-12)

• (x+12)(x+12)

• Not Factorable

Multiple Select

Check all the expressions that are difference of two squares.

• $$x^2-25$$

• $$x^2-100$$

• $$x^2+36$$

• $$x^2-73$$

• $$16x^2-400$$

Open Ended

Why is

$$x^2-73$$ not difference of two squares? 

Fill in the Blank

Factor

$$x^2-121$$  

Fill in the Blank

$$c^2-81d^2$$  Type your answer with parentheses and NO SPACES.

Fill in the Blank

Factor the following completely:

$$45q^3\ -\ 20q$$