How to Factor

What is a Quadratic Expression

A quadratic expression is an expression where the largest exponent for a variable is 2.

Ex: 2

2

2

2

4 33

2 4 7

4

6 3

x

x x

x

x

x

x

What is a Quadratic Expression

A quadratic expression can come in different forms.

Form General Form Example

Standard Form

Factored Form

2ax bx c

( )( )x r x s ( 3)( 5)x x

2 53 2xx

Standard Form

• a, b & c are numbers that are called “coefficients”

2ax bx c

2 53 2xx

3a 2b5c

Factored Form

• r & s are numbers

( )( )x r x s

( 3)( 5)x x

3r 2s

What is Factoring

Factoring is the process for turning a “standard” form expression into a “factored” form expression

( )( )x r x s 2ax bx c

What is Factoring

A quadratic expression in standard form can also sometimes be called a “trinomial” when it has three terms

2 4 5x x term #1 term #2 term #3

1a 4b 5c

4 Types of Trinomials

1. Simple Trinomial

2. Simple Trinomial with Common Factoring

3. Complex Trinomial

4. Complex Trinomial with Common Factoring

Simple Trinomial - (a=1)

A trinomial who’s “a” value is 1

2 4 5x x 1a

Simple Trinomial with Common Factoring

A trinomial who’s “a” value is 1 after a common factor is removed.

2 8 102x x

2 4 )( 52 x x

Now a simple trinomial

By common factoring

Complex Trinomial - (a≠1)

A trinomial who’s “a” value is NOT 1

23 4 5x x 3a

Complex Trinomial with Common Factoring

A trinomial who’s “a” value is NOT 1 after a common factor is removed.

2 8 104x x

2(2 5)2 4x x

Now a complex trinomial

By common factoring

Which is Which?

What types of trinomial is below:

23 8 9x x

2 30 909x x

2 17 92x x 2 344x x 2 1x x

2 12 273x x 2 4 9x x

25 5 5x x

2 898752 1237x x

Factoring a Simple Trinomial

2 5 6x x 6c

1. Write out the factors of the “c” in pairs.

6 : (1)(6)(2)(3)

2 3 5

6c

2 5 6 ( 2)( 3)x x xx

Factored!

2. Now chose a pair whose numbers can be added or subtracted to make the “b” value

3. Use these two numbers to make the factors of the expression

Factoring a Simple Trinomial

2 3 4x x 1. Write out the factors of the “c” in pairs.

4 : (1)(4)

41 4 3

2 3 4 ( )( )x xx x

c b2. Now chose a pair whose numbers can be

added or subtracted to make the “b” value

3. Use these two numbers to make the factors of the expression

3

(2)(2)

1 4 3

( )(1 4)x x

Factoring a Simple Trinomial

2 6x x 1. Write out the factors of the “c” in pairs.

6 : (1)(6)

62 3 3

2 ( )( )6 xx x x

c b2. Now chose a pair whose numbers can be

added or subtracted to make the “b” value

3. Use these two numbers to make the factors of the expression

1(2)(3)

2 3 1

( )(2 3)x x

Factoring a Complex Trinomial

2 25 122x x 2. Write out the factors of the “ac” in pairs.

(1)24 : (24)24

1 24 25

ac b

3. Now chose a pair whose numbers can be added or subtracted to make the “b” value

25 (3)(8)

1 24 25

1. Multiply the “a” and “c” values.

2(12) 24

(4)(6)(2)(12)

Factoring a Complex Trinomial

2 25 122x x 1 24 25

3. Now chose a pair whose numbers can be added or subtracted to make the “b” value

1 24 25 1 24x x4. Split the “bx” into two terms using the

factor pairs from above.2 22 24 1x xx 5. Common factor the first two terms and

then factor the second two terms.(2 1) 12(2 1)x x x

6. Place the two terms multiplying the brackets into a factor

7. The other factor is the terms inside the brackets

( 12)x (2 1)x