How to Factor
What is a Quadratic Expression
A quadratic expression is an expression where
the largest exponent for a variable is 2.
Ex: 3 x 2  4 x  3
2 x 2  4 x  7
x 4
2
x  6x2  3
What is a Quadratic Expression
A quadratic expression can come in different
forms.
Form
General Form
Example
Standard Form
ax 2  bx  c
3x 2  2x  5
Factored Form
( x  r )( x  s )
( x  3)( x  5)
Standard Form
ax  bx  c
2
• a, b & c are numbers that are called
“coefficients”
3x  2x  5
2
a  3
b2
c5
Factored Form
( x  r )( x  s )
• r & s are numbers
( x  3)( x  5)
r  3
s2
What is Factoring
Factoring is the process for turning a “standard”
form expression into a “factored” form expression
ax  bx  c
2
( x  r )( x  s )
What is Factoring
A quadratic expression in standard form can also
sometimes be called a “trinomial” when it has
three terms
x  4x  5
2
term #1
a 1
term #2
b4
term #3
c5
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
x  4x  5
2
a 1
Simple Trinomial with Common Factoring
A trinomial who’s “a” value is 1 after a
common factor is removed.
2x  8 x  10
2
By common factoring
2( x  4 x  5)
2
Now a simple trinomial
Complex Trinomial - (a≠1)
A trinomial who’s “a” value is NOT 1
3x  4 x  5
2
a3
Complex Trinomial with Common Factoring
A trinomial who’s “a” value is NOT 1 after a
common factor is removed.
4x  8 x  10
2
By common factoring
2(2 x  4 x  5)
2
Now a complex trinomial
Which is Which?
What types of trinomial is below:
3x  8 x  9
x  4x  9
3x  12 x  27
2x  17 x  9
x  4 x  34
x  x 1
2
2
2
2
9x  30 x  90
2
2
5x  5x  5
2
x  898752 x  1237
2
2
Factoring a Simple Trinomial
x  5x  6
2
c6
1. Write out the factors of the “c” in pairs.
c6
6 : (1)(6)
(2)(3)
2. Now chose a pair whose numbers can be
added or subtracted to make the “b” value
23 5
3. Use these two numbers to make the
factors of the expression
x 2  5 x  6  ( x  2)( x  3)
Factored!
Factoring a Simple Trinomial
x  3x  4
2
c
4
b
3
x 2  3 x  4  ( x  )( x  )
( x  1)( x  4)
1. Write out the factors of the “c” in pairs.
4 : (1)(4)
(2)(2)
2. Now chose a pair whose numbers can be
added or subtracted to make the “b” value
1 43
1  4  3
3. Use these two numbers to make the
factors of the expression
Factoring a Simple Trinomial
x  x6
2
c
6
b
1
x 2  x  6  ( x  )( x  )
( x  2)( x  3)
1. Write out the factors of the “c” in pairs.
6 : (1)(6)
(2)(3)
2. Now chose a pair whose numbers can be
added or subtracted to make the “b” value
2 33
2  3  1
3. Use these two numbers to make the
factors of the expression
Factoring a Complex Trinomial
2x  25 x  12
2
ac
24
b
25
1. Multiply the “a” and “c” values.
2(12)  24
2. Write out the factors of the “ac” in pairs.
24 : (1)(24)
(2)(12)
(3)(8)
(4)(6)
3. Now chose a pair whose numbers can be
added or subtracted to make the “b” value
1 24  25
1  24  25
Factoring a Complex Trinomial
2x  25 x  12
2
1x  24 x
3. Now chose a pair whose numbers can be
added or subtracted to make the “b” value
1 24  25
1  24  25
4. Split the “bx” into two terms using the
factor pairs from above.
 2 x 2  x  24 x  12
 x(2 x  1)  12(2 x  1)
 ( x  12) (2 x  1)
5. Common factor the first two terms and
then factor the second two terms.
6. Place the two terms multiplying the
brackets into a factor
7. The other factor is the terms inside the
brackets