Factoring

Factoring by Greatest Common Factor (GCF)

Factoring by removing the GCF “undoes” the multiplication. (It’s the distributive property in reverse.)

2x2 – 4

2x2 – (2 2)

2(x2 – 2)

Check 2(x2 – 2)

2x2 – 4

Example

8x3 – 4x2 – 16x

2x2(4x) – x(4x) – 4(4x)

4x(2x2 – x – 4)

4x(2x2 – x – 4)

8x3 – 4x2 – 16x

Check

Factoring ax2+bx+c

2x bx c

To factor, find two numbers that:

Add to b Multiply To c

Example

Factor x2 + 15x + 36. Check your answer.

(x + )(x + )

(x + 1)(x + 36) = x2 + 37x + 36

(x + 2)(x + 18) = x2 + 20x + 36

(x + 3)(x + 12) = x2 + 15x + 36

Example

Factor each trinomial. Check your answer.

x2 + 10x + 24

(x + )(x + )

(x + 4)(x + 6) = x2 + 10x + 24

Factor each trinomial.

x2 + x – 20

(x + )(x + )

(x – 4)(x + 5)

Remember to look for a GCF first!

3x2 + 18x – 21

3(x + )(x + )

3(x – 1)(x + 7)

3(x2 + 6x – 7)

Factor each trinomial.

x2 – 11xy + 30y2

(x – 5y)(x – 6y)

A trinomial is a perfect square if:

• The first and last terms are perfect squares.

• The middle term is double the product of

the square roots.

9x2 + 12x + 4

3x 3x 2(3x 2) 2 2 • • •

REMEMBER THE X!

Perfect Square Trinomials

Factor.

81x2 + 90x + 25

Example

(9x)( 9x) (5)( 5)

The middle term = 2(9x)(5), so this is a perfect square trinomial

Factor.

Example

281 90 25

(9 )(9 )

x x

x x

Factor.

Example

281 90 25

(9 5 )(9 5)

x x

x x

A polynomial is a difference of two squares if:

•There are two terms, one subtracted from the other.

• Both terms are perfect squares.

4x2 – 9

2x 2x 3 3

For variables: All even powers are perfect squares

Difference of Two Squares

Example

249 100x

(7x)( 7x) (10)( 10)

Example

249 100

(7 10)(7 10)

x

x x

Center term cancels out, so use opposite signs.

Factoring ax2 + bx + c

The “box” method

22x

6

First term

Last term

Include the signs!!!

22 7 6x x

Multiply a and c. (In this case, that would be 2 x 6 = 12) Put the

factors of “ac” that add up to “b” in the other squares, with their signs and an “x”. Order does not matter.

22x

6

One factor

The other factor

3x

4x

22 7 6x x

4 and 3 are factors of 12 that add up to 7, so they go in the empty spaces. Add an x because they represent the middle term.

Factor the GCF out of each row or

column. Use the signs from the closest term.

22x

63x

4xGCF is +2x

Factor the GCF out of each row or

column. Use the signs from the closest term.

22x

63x

4x+2x

GCF is +3

Factor the GCF out of each row or

column. Use the signs from the closest term.

22x

63x

4x

GCF is +x

+2x

+3

Factor the GCF out of each row or

column. Use the signs from the closest term.

22x

63x

4x

+x GCF =+2

+2x

+3

The “outside” factors combine to factor the quadratic.

22x

63x

4x

+x +2

+2x

+3

(2x+3)(x+2)

You can also use trial and error. Determine the possible factors for the first and last terms, and then keep trying combinations until you find the

one that works.

23 17 10 x x

First term: 3x and x are the only possible factors Last term: Factors are 1 and 10 or 2 and 5

2(3 1)( 10) 3 31 10 x x x x

2(3 10)( 1) 3 13 10 x x x x

2(3 5)( 2) 3 11 10 x x x x

2(3 2)( 5) 3 17 10 x x x x

Only this combination works.

Factoring by Grouping

Factoring by grouping: If you have four terms – make 2 groups of 2 and factor out the GCF from each. MUST be used on 4 terms CAN be used on 3.

Example

Factor each polynomial by grouping. Check your answer.

6h4 – 4h3 + 12h – 8

(6h4 – 4h3) + (12h – 8)

2h3(3h – 2) + 4(3h – 2)

2h3(3h – 2) + 4(3h – 2)

(3h – 2)(2h3 + 4)

Factor each polynomial by grouping. Check your answer.

Check (3h – 2)(2h3 + 4)

3h(2h3) + 3h(4) – 2(2h3) – 2(4)

6h4 + 12h – 4h3 – 8

6h4 – 4h3 + 12h – 8

Example

Factor each polynomial by grouping. Check your answer.

5y4 – 15y3 + y2 – 3y

(5y4 – 15y3) + (y2 – 3y)

5y3(y – 3) + y(y – 3)

5y3(y – 3) + y(y – 3)

(y – 3)(5y3 + y)

Factor each polynomial by grouping. Check your answer.

5y4 – 15y3 + y2 – 3y

Check (y – 3)(5y3 + y)

y(5y3) + y(y) – 3(5y3) – 3(y)

5y4 + y2 – 15y3 – 3y

5y4 – 15y3 + y2 – 3y

You can use factoring by grouping on trinomials.

23 11 10x x

Split the 11x into two terms (coefficients should multiply to 30, because 3x10=30)

23 6 5 10x x x

Write the terms in whichever order will allow you to group.

2(3 6 ) (5 10)

3 ( 2) 5( 2)

(3 5)( 2)

x x x

x x x

x x