FACTORING BINOMIALS of the form a b
2 2
Examples of “Perfect Squares”
4 is a perfect square because 4 = (2)2
9 is a perfect square because 9 = (3)2
1 is a perfect square because 1 = (1)2
x2 is a perfect square because x2 = (x)2
y4 is a perfect square because y4 = (y2)2
z6 is a perfect square because z6 = (z3)2
16x2 is a perfect square because 16x2 = (4x)2
25y4z10 is a perfect square because 25y4z10 = (5y2z5)2
Difference of Two Squares
a2 b2 a b a b
Step 1. Are there any common factors? If so, factor out the GCF.
Step 2. Write each term of the binomial as a quantity squared if
possible.
For example: The term 4x 2 can to written as 2x .
2
Example 1.
Factor x 2 100
x 10
2 2
a2 b 2 a b a b
a x and b 10
x 10 x 10
1
Example 2.
Factor 4 x 2 16
2x 4
2 2
a2 b 2 a b a b
a 2 x and b 4
2x 4 2x 4
Example 3.
Factor 16 x 2 9 y 2
4x 3 y
2 2
a2 b 2 a b a b
a 4 x and b 3 y
4x 3 y 4x 3 y
Example 4.
Factor 36 yz 5 9 yz 3
9 yz 3 4 z 2 1
9 yz 3 2 z 1
2 2
a 2 b 2 a b a b
a 2 z and b 1
9 yz 3 2 z 1 2 z 1
Example 5.
Factor 9 x 2 16 y 2
Prime. You can not factor the SUM of 2 squares.
2
FACTORING PERFECT SQUARE TRINOMIALS
Perfect Square Trinomials
a 2 2ab b2 a b a b a b
2
a 2 2ab b2 a b a b a b
2
Example 1.
x 2 18 x 81
x 2 x 9 9
2 2
a 2 2ab b2 a b
2
a x and b 9
x 9
2
Note: You could also factor this as a trinomial of the form x 2 bx c . Find
two factors of 81 that add to 18.
Example 2.
4 x 2 y 2 28 xy 49
2 xy 2 2 xy 7 7
2 2
2ab b2 a b
2
a2
a 2 xy and b 7
2 xy 7
2
Example 3.
36 p 2 36 pq 9q 2
6 p 2 6 p 3q 3q
2 2
2ab b2 a b
2
a2
a 6 p and b 3q
6 p 3q
2
3