# Factoring Polynomials

**Types of Factoring**

1) Factor by taking out a Greatest Common Factor (GFC)

2) Factor a trinomial as two binomials .

3) Factor a binomial as a difference of two squares

4) Factor a binomial as a difference of cubes or a sum of cubes

**Factoring out a Greatest Common Factor**

Taking out a greatest common factor is essentially the same as working the
distributive

property backwards .

Review of distributive property

3(x + 5) = 3x + 3·5 = 3x +15

**Example 1**

Factor out a greatest common factor 4x +16

Solution : 4x +16 = 4(x + 4)

**Example 2**

Factor out a greatest common factor 5x − 35

Solution: 5x − 35 = 5(x − 7)

**Example 3**

Factor out a greatest common factor

Solution:

**Example 4**

Factor out a greatest common factor

Solution:

**Factoring a trinomial as two binomials**

Factoring a trinomial is the same as working the **FOIL** process

So, here is a short review of** FOIL**

Example of factoring a trinomial as two binomials

Factor x^{2} +10x + 25

In this example you want to find two numbers that multiply to get 25 and add to
get 10.

By using x as the first entry in each binomial you get:

x^{2} +10x + 25 = (x + 5)(x + 5)

Here are some similar examples

**Example 5**

Factor x^{2} −10x + 24

Answer: (x − 6)(x − 4) Hint: Find two numbers that multiply to get 24 and add to
get -

10

**Example 6**

Factor x^{2} − 2x − 35

Answer: (x − 7)(x + 5) Hint: Find two integers that multiply two get -35 and add

to get -2 which is the factors -7 and 5

**Example 7**

Factor x^{2} + 4x −12

Answer: (x − 2)(x + 6)

**Example 8**

Factor 3x^{2} + 7x + 2

Answer: (3x +1)(x + 2)

**Factor a binomial as a difference of squares **

Factor x^{2} − 4

Answer: (x − 2)(x + 2) Hint: Basically use the **FOIL** process backwards again and

find two integers that multiply to get -4 and add to get zero . This process will
cancel out

the x- terms .

Check:

Other similar examples

**Example 9**

Factor m^{2} − 36

Answer: (x − 6)(x + 6)

**Example 10**

Factor x^{2} −144

Answer: (x +12)(x −12)

**Example 11**

Factor 9x^{2} − 25y^{2}

Answer:

**General Form of a Difference of Squares: **A^{2} − B^{2} = (A + B)(A − B)

**Difference of Cubes and sum of Cubes
Main formulas
Difference of Two Cubes**

A

^{3}− B

^{3}= (A − B)(A

^{2}+ AB + B

^{2})

**Sum of Two Cubes**

A

^{3}+ B

^{3}= (A + B)(A

^{2}− AB + B

^{2})

**Example 12**

Factor x^{3} − 27

Answer:

**Example 13**

Factor x^{3} + 64

Answer:

**Example 14**

Factor x^{3} + 8

Answer;

**Example 15**

Factor 8x^{3} − 27y^{3}

Answer;

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