# 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}