MATH 0308 / MATH 0310
Factoring Polynomials
Factoring the GCF
Factoring by Patterns
(Special Products)
Difference of Squares
Sum of Cubes
The Greatest Common Factor (GCF) is the largest factor that divides into each term of a
polynomial.
Example:
3x2+ 6xy + 9x = 3x(x + 2y + 3)
GCF = 3x
Difference of Squares
Sum of Squares
Difference of Cubes
Sum of Cubes
Trinomial Squares
x2 y 2 ( x
Example:
y )( x y )
4x2 16
x2
y2
(x
x
2
y )( x
y)
2
y = prime, cannot be factored
x3
y3
(x
y )( x 2
xy
y2 )
x3
y3
(x
y )( x 2
xy
y2 )
x2
2 xy
y2
(x
y)2
x2
2 xy
y2
(x
y)2
= 4(x2 4)
= 4(x 2)(x + 2)
x3 + y3 = (x + y)(x2 xy + y2)
Example:
x3 + 27
= (x3 + 33)
= (x + 3)(x2
Difference of Cubes
x3 – y3 = (x – y)(x2 + xy + y2)
Example:
x3 – 8
= x3 – 23
= (x-2)(x2 + 2x + 4)
Trinomial Squares
x 2 2 xy y 2 ( x y ) 2 or x 2
Example:
x2 – 6x + 9
2 xy
y2
GCF = 4
3x + 9)
(x
The first and third terms are positive
The first and third terms are perfect squares: x2 and 32
Twice their product is the middle term: 2 (x 3) = 6x Disregard the sign
Therefore, x2 – 6x + 9 = (x – 3)2
Use the sign of the middle term
Example:
9x2 + 6x + 1
The first and third terms are positive
The first and third terms are perfect squares: (3x)2 and 12
Twice their product is the middle term: 2 (3x 1) = 6x
Therefore, 9x2 + 6x + 1 = (3x + 1)2 Use the sign of the middle term
y)2
Factoring by Grouping
This method is used when there are four (4) terms in the polynomial.
Example: 3xy – 6x + 2y – 4
= (3xy – 6x) + (2y – 4)
group the terms into pairs
= 3x(y – 2) + 2(y – 2)
take out the GCF from each pair
= (y – 2)( 3x + 2)
take out the common binomial factor
Factoring a Trinomial by the Reverse FOIL Method
Consider multiplying ( F1 L1 )( F2
L2 ) :
F = First = F1 F2 , O = Outer = F1 L2 , I = Inner = L1 F2 , L = Last = L1 L2
Examples
( x 1)( x 2)
( x 3)( x 4)
(2 x 1)( x 5)
(3x 2)( x 3)
=
=
=
=
F
x2
x2
x2
x2
O
2x
4x
-10x
-9x
I
1x
-3x
1x
-2x
L
2
-12
-5
6
=
=
=
=
Result
x + 3x + 2
x 2 + x 12
x 2 9x 5
x 2 11x+ 6
2
The Reverse FOIL Method is simply transforming the Result column back to the first column.
In other words, factoring a trinomial means writing it as a product of factors.
The Result column is in the form of a trinomial, ax 2 bx c . Notice that:
the first term equals the value in column F
the middle term equals O + I
the last term equals L
Steps:
1) Factor the first term into the “F” positions: ( F1
2) Factor the last term into the “L” positions: (
) ( F2
L1 ) (
).
L2 ).
3) Check the factoring by FOIL. Make sure Outer and Inner add up to the middle term. If not, try switching
factors and signs inside the parentheses until the FOIL result equals the trinomial.
Factoring a Trinomial by the ac-Method
Steps:
1) Find the value of a c
2) Look for factors of ac that will add to b
3) Split the middle term into two terms with the result in step (2)
4) Factor by grouping
3x 2 10x 8
Example:
a = 3, b = -10, c = -8
1. (3)(-8) = -24
2. Look for factors of -24 that will add to -10
-24 = 2 -12
2 + (-12) = -10
2
2
3. 3 x 10 x 8 = 3x 2 x 12 x 8
4. 3x2 2 x 12 x 8 = x(3x + 2) – 4(3x + 2) = (3x + 2)(x – 4)
LSC-Montgomery Learning Center: Factoring Polynomials
Last Updated April 13, 2011
Page 2
Flowchart for Factoring a Polynomial
GREATEST COMMON FACTOR (GCF)?
YES
NO
HOW MANY TERMS?
FACTOR GCF
2
4
3
BINOMIAL (2)
POLYNOMIAL (4)
TRINOMIAL (3)
ax2+bx+c
FACTOR BY
GROUPING
YES
TRINOMIAL
SQUARE?
FACTOR BY
PATTERNS
NO
YES
FACTOR BY
PATTERNS
x2-y2=(x+y)(x-y)
x -y3=(x-y)(x2+xy+y2)
x3+y3=(x+y)(x2-xy+y2)
x2+2xy+y2=(x+y)2
x2-2xy+y2=(x-y)2
3
a=1
a>1
Reverse
FOIL
(Trial & Error)
Reverse FOIL
or
ac-METHOD
CANNOT BE
FACTORED
LSC-Montgomery Learning Center: Factoring Polynomials
Last Updated April 13, 2011
Page 3