Polynomial factoring review

Polynomial multistep factoring review

ALL the ways to factor: GCF, trinomial grouping, difference of perfect squares, difference of perfect cubes, and higher degree polynomial grouping

Greatest Common Factor (GCF)

The highest expression that divides into the terms.

This should always be your first step, since it will start with simplest form and can make it possible to use other methods. From there, strategize about which method will be most helpful.

Multiple Choice

What is the first question you ask yourself when factoring a polynomial? 

• How many terms does it have? 
• Is there a GCF? 
• What strategy should I use? 
• Why do I have to learn this?!?!?!

Fill in the Blank

What is the GCF of the following?

$$12x^3+9yx$$  

Multiple Choice

Factor the common factor out of each expression.

-15m⁶ - 30m⁴ - 18m³

• -3m² (5m³ + 10m + 6)

• -3m⁴ (m³ + 5m + 6)

• -3m³ (5m⁴ + 10m + 6)

• -3m³ (5m³ +10m + 6)

Trinomial factor by grouping

Multiple Choice

Factor the expression: 10v² + 11v - 8

• (5v + 8)(2v + 1)

• (v + 8)(v - 1)

• (5v + 8)(2v - 1)

• (10v - 8)(v + 1)

Multiple Choice

Factor completely.
3x² - 9x -120

• 3(x - 8)(x + 5)
• 3(x + 8)(x - 5)
• (3x + 8)(x - 5)
• (3x - 8)(x + 5)

Difference of perfect squares

Remember: this ONLY works if you have subtraction.

When you have an expression with only two terms, always look to see if there is a something you can factor out to make them perfect squares or perfect cubes.

Double difference of perfect squares

If it is a higher degree polynomial, check your factors for ANOTHER difference of perfect squares.

You may have to do perfect squares TWICE to factor completely.

Multiple Choice

Factor 25x² - 81.

• (5x -  9)(5x + 9)
• (5x - 9)(5x - 9)
• (5x + 9)(5x + 9)
• cannot be factored

Multiple Choice

Factor Completely:
4x² + 64

• 4(x² + 16)
• 4(x - 4)(x + 4)
• 4(x² + 64)
• Prime

Multiple Choice

Factor Completely: 16x⁴-1

• (2x+1)(2x-1)(4x²+1)

• (4x²+1)(4x²-1)

• 4x²(4x²-1)

• (2x+1)(2x-1)(2x+1)(2x-1)

Sum or difference of perfect CUBES

Follow the pattern, and remember to use SOAP!

When you have an expression with only two terms, always look to see if there is a something you can factor out to make them perfect squares or perfect cubes.

Multiple Choice

Factor: 27x³ + 8

• (3x + 2)(9x² - 6x + 4)

• (9x + 2)(3x² + 18x + 4)

• (3x²+4)(9x + 2)

• (27x + 8)(x² - 216x + 64)

Multiple Choice

Factor $$27x^3-343$$  

• $$\left(3x+7\right)\left(3x^2+21x+49\right)$$  

• $$\left(3x+7\right)\left(9x^2-21x+14\right)$$  

• $$\left(3x-7\right)\left(3x^2+10x+49\right)$$  

• $$\left(3x-7\right)\left(9x^2+21x+49\right)$$  

Higher degree factor by grouping

Factoring by grouping also works with four-term polynomials of higher degree.

Look to see if you can factor out a GCF first, and double check that your factors aren't perfect squares that need to be factored again. 

Multiple Choice

x³-5x²+5x-25

• (x²-5)(x-5)
• (x²+5)(x-5)
• (x²+5)(x+5)
• (x²-1)(x-25)

Multiple Choice

30x³+15x²-18x-9

• (5x²-3)(6x+1)
• (5x²-3)(6x+3)
• 3(5x²+3)(2x-1)
• 3(5x²-3)(2x+1)

Multiple Choice

Factor completely:

$$-6u^5-14u^4+6u^3+14u^2$$  

• $$-2u^2\left(3u^3+7u^2-3u-7\right)$$  

• $$-2u^2\left(3u+7\right)\left(u^2-1\right)$$  

• $$-2u^2\left(3u+7\right)\left(u-1\right)\left(u+1\right)$$  

Match

Match the following: Factor each and match.

• (x-4)(x+7)

• (x - 11)(x + 3)

• (2x - 5)(x + 8)

• (2x + 1)(x - 7)

• 2(x + 5)(x - 5)

x² +3x -28

x² - 8x - 33

2x² + 11x -40

2x² - 13x - 7

2x² - 50

• x² +3x -28

• x² - 8x - 33

• 2x² + 11x -40

• 2x² - 13x - 7

• 2x² - 50