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
a^{2} + 2ab + b^{2} = (a + b)^{2}
a^{2}  2ab + b^{2} = (a  b)^{2}
Consider the polynomial x^{2} + 6x + 9. If we recognize that
x^{2} + 6x + 9 = x^{2} + 2 Â· x Â· 3 + 3^{2},
then we can see that it is a perfect square trinomial. It fits the rule if a
= x and b = 3:
x^{2} + 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 a^{2} and b^{2},
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) x^{2}  8x + 16
b) a^{2} + 14a + 49
c) 4x^{2} + 12x + 9
Solution
a) Because the first term is x^{2}, the last is 4^{2}, and 2(x)(4) is equal to the middle term
8x, the trinomial x^{2}  8x + 16 is a perfect square trinomial:
x^{2}  8x + 16 = (x  4)^{2}
b) Because 49 = 7^{2} and 14a = 2(a)(7), we have a perfect square trinomial:
a^{2} + 14a + 49 = (a + 7)^{2}
c) Because 4x^{2} = (2x)^{2}, 9 = 3^{2}, and the middle term 12x is equal to 2(2x)(3), the
trinomial 4x^{2} + 12x + 9 is a perfect square trinomial:
4x^{2} + 12x + 9 = (2x + 3)^{2}
