A2H Unit 9: Factoring Polynomials

Unit 9: Polynomials

Topic #1: Factoring Quadratics Review

• trinomial where a = 1

• trinomial where a > 1 (grouping)

• difference of squares

Topic #2: Factoring Cubic Expressions

• grouping

• difference of cubes

• sum of cubes

• Soon, we are going to be analyzing the graphs of polynomial functions and looking for specific features.

• Factoring specifically shows us the "zeros" of the functions.

What are we going to use this for?

​This video covers the following types of factoring:

• GCF

• trinomials when a = 1

• trinomials when a > 1

• difference of squares

• grouping w./ cubics

Multiple Choice

Factor completely:
16x² - 20xy

• 4 (4x² - 5xy)

• 4x (4x - 5y)

• 4x² (4 - 5y)

• 2(duck)

Multiple Choice

Factor by grouping:
x³+7x²-2x-14

• (x²-2)(x-7)

• (x²+2)(x-7)

• (x²+2)(x+7)

• (x²-2)(x+7)

Multiple Choice

Write x³ + 9x² + 18x in factored form

• x(x + 6)(x + 3)

• x(x + 9)(x + 2)

• x(x + 6)

• x(x - 6)(x - 3)

Multiple Select

Using factoring, select all zeros of: $$\left(x^2-1\right)\left(x^{2\ }+x-12\right)$$   HINT: Factor both of the polynomials

• 1

• -1

• -4

• 3

• -3

Multiple Choice

Which expression is a factor of $$121x^2-36$$ ?

• 11x+6

• 11x+12

• 11x-36

• 121x-6

Math Response

What are the factors of $$x^2-49$$ ? (put each factor in parentheses with no spaces)

Type answer here

Correct Answer: \left(x-7\right)\left(x+7\right)

Deg^°

Rad

Multiple Choice

Find the factors of

x² + x - 6

• (x - 6)(x - 9)

• (x + 3)(x + 2)

• (x + 3)(x - 2)

• (x + 2)(x + 4)

Multiple Choice

Find the factors of

x² - 5x - 36

• (x + 4)(x - 9)

• (x - 4)(x + 9)

• (x - 9)(x + 6)

• (x + 4)(x + 9)

Multiple Choice

Find the factors of

5m² - 45m + 90

• 3(m + 1)(m + 8)

• 5(m - 3)(m - 6)

• 2(m - 7)(m + 3)

• 5(m - 5)(m + 3)

Multiple Choice

Factor Completely.  
8x³ - 18x

• 2x(4x²-9)                                    

• (4x²+3)(2x-3)

• 2x(2x+3)(2x-3)

• PRIME

Multiple Choice

Factor by grouping:
3x³-x²+6x-2

• (x²+2)(3x+1)

•  (x²+2)(3x-1)

• (3x³+6x)(x²-2)

• none of the aboe

Multiple Choice

4x² - 64y²

• Prime

• ( 2x - 32y) (2 x + 32y) 

• ( 2x - 8y) (2 x - 8y) 

• ( 2x - 8y) (2 x + 8y) 

Multiple Choice

Factor the GCF
4b⁵+4b³+16b²

• 4b³(b⁴+b²+4b)

• 4b³(b⁴+4b+5)

• 4(b⁵+b³+4b²)

• 4b²(b³+b+4)

Multiple Choice

What are the remaining factors for

(x³ +x² -17x +15) if (x -1) is a factor? (use synthetic division)

• (x-1) (x+5) (x-3)

• (x-1) (x+5) (x + 3)

• (x+1) (x+5) (x-3)

• (x+1) (x-5) (x-3)

​This video covers more factoring of cubic expressions. If you struggled with any of the previous questions, this video might help you understand the connection between synthetic division and factoring more.

Multiple Choice

            Is (x-2) a factor of             f(x)= x³-8x²+14x-4?

• Yes, (x-2) is a factor. There is a remainder.

• No, (x-2) is  not a factor. The remainder is zero.

• Yes, (x-2) is a factor. The remainder is zero.

• No, (x-2) is  not a factor. There is a remainder. 

Multiple Choice

One of the factors of the polynomial below is x-4. Write the polynomial in its fully factored form.

$$f\left(x\right)=2x^3+7x^2-53x-28$$  

• $$f\left(x\right)=\left(x-4\right)\left(2x+1\right)\left(x+7\right)$$  

• $$f\left(x\right)=\left(x-4\right)\left(2x^2+15x+7\right)$$  

• $$f\left(x\right)=2x^2+15x+7$$  

• $$f\left(x\right)=\left(2x+1\right)\left(x+7\right)$$  

​This video covers sums and differences of cubic expressions. On the next slide, I'll have the formulas that he shows on his whiteboard.

Factoring a Sum or Difference of Cubes

Multiple Choice

Complete the formula
a³-b³=

• (a-b)(a²+ab+b²)

• (a+b)(a²-ab+b²)

• (a-b)(a²-ab+b²)

• (a+b)(a²+ab+b²)

Multiple Choice

Complete the formula

a³+b³=

• (a-b)(a²+ab+b²)

• (a+b)(a²-ab+b²)

• (a-b)(a²-ab+b²)

• (a+b)(a²+ab+b²)

Multiple Choice

Factor:   x³ + 1

• (x + 1)³

• (x+1)(x² - x + 1)

• (x - 1) (x² + x - 1)

• (x + 1)(x² - 2x + 1)

Multiple Choice

Factor: 8x³ - 1

• (2x - 1)(4x² + 2x + 1)

• (2x + 1)(4x² - 2x + 1)

• (2x - 1)(4x² - 2x - 1)

• (2x + 1)(4x² +2x + 1)

Multiple Choice

Factor
27x³-64

• (3x-4)(9x²+12x+16)

• (3x-4)(9x²-12x+16)

• (3x+4)(9x²-12x+16)

• (3x+4)(9x²+12x+16)

Multiple Choice

Fully Factor the Following
8x³-27

• (2x+3)(4x²-6x+9)

• (2x-3)(4x²+6x+9)

• (2x-3)(4x²+6x-9)

• (2x-3)(2x²+6x+9)

Multiple Choice

Which number is both a perfect square and a perfect cube?

• 64

• 27

• 81

• 16

Multiple Choice

Identify if the expression can be factored as sum or difference of cubes:

3x³ + 24

• No

• Yes

• Yes, getting GCF first

• It is a sum of squares