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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


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

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