Factoring Perfect Square Trinomials
The trinomial that results from squaring a binomial is called a perfect square
trinomial. We can reverse the rules from Section 5.4 for the square of a sum or a
difference to get rules for factoring.
Factoring Perfect Square Trinomials
a2 + 2ab + b2 = (a + b)2
a2 - 2ab + b2 = (a - b)2
Consider the polynomial x2 + 6x + 9. If we recognize that
x2 + 6x + 9 = x2 + 2 · x · 3 + 32,
then we can see that it is a perfect square trinomial. It fits the rule if a
= x and b = 3:
x2 + 6x + 9 = (x - 3)2
Perfect square trinomials can be identified by using the following strategy.
Strategy for Identifying Perfect Square Trinomials
A trinomial is a perfect square trinomial if
1. the first and last terms are of the form a2 and b2,
2. the middle term is 2 or -2 times the product of a and b.
We use this strategy in the next example.
Example 1
Factoring perfect square trinomials
Factor each polynomial.
a) x2 - 8x + 16
b) a2 + 14a + 49
c) 4x2 + 12x + 9
Solution
a) Because the first term is x2, the last is 42, and -2(x)(4) is equal to the middle term
-8x, the trinomial x2 - 8x + 16 is a perfect square trinomial:
x2 - 8x + 16 = (x - 4)2
b) Because 49 = 72 and 14a = 2(a)(7), we have a perfect square trinomial:
a2 + 14a + 49 = (a + 7)2
c) Because 4x2 = (2x)2, 9 = 32, and the middle term 12x is equal to 2(2x)(3), the
trinomial 4x2 + 12x + 9 is a perfect square trinomial:
4x2 + 12x + 9 = (2x + 3)2 |
