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 x² +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:

x² +10x + 25 = (x + 5)(x + 5)

Here are some similar examples

Example 5

Factor x² −10x + 24

Answer: (x − 6)(x − 4) Hint: Find two numbers that multiply to get 24 and add to get -
10

Example 6

Factor x² − 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 x² + 4x −12

Answer: (x − 2)(x + 6)

Example 8

Factor 3x² + 7x + 2

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

Factor a binomial as a difference of squares

Factor x² − 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 m² − 36

Answer: (x − 6)(x + 6)

Example 10

Factor x² −144

Answer: (x +12)(x −12)

Example 11

Factor 9x² − 25y²

Answer:

General Form of a Difference of Squares: A² − B² = (A + B)(A − B)

Difference of Cubes and sum of Cubes

Main formulas

Difference of Two Cubes
A³ − B³ = (A − B)(A² + AB + B² )

Sum of Two Cubes
A³ + B³ = (A + B)(A² − AB + B² )

Example 12

Factor x³ − 27

Answer:

Example 13

Factor x³ + 64

Answer:

Example 14

Factor x³ + 8

Answer;

Example 15

Factor 8x³ − 27y³

Answer;