Beginning Algebra, 11th Edition

On the one hand, a necessary condition for a trinomial to be a perfect square is

that two of its terms be perfect squares.For this reason, is not a per-

fect square trinomial, because only the term is a perfect square.

On the other hand, even if two of the terms are perfect squares, the trinomial may

not be a perfect square trinomial. For example, has two perfect square

terms, x^2 and 36, but it is not a perfect square trinomial.

x^2 + 6 x+ 36

16 x^2

16 x^2 + 4 x+ 15

SECTION 6.4 Special Factoring Techniques 383

NOW TRY
EXERCISE 4
Factor .y^2 + 14 y+ 49

Factoring Perfect Square Trinomials

x^2  2 xyy^2  1 xy 22

x^2  2 xyy^2  1 xy 22

The middle term of a perfect square trinomial is always twice the product of the

two terms in the squared binomial (as shown in Section 5.6). Use this rule to check

any attempt to factor a trinomial that appears to be a perfect square.

Factoring a Perfect Square Trinomial

Factor

The -term is a perfect square, and so is 25.

Try to factor as

To check, take twice the product of the two terms in the squared binomial.

x^2 + 10 x+ 25 1 x+ 522.

x^2

x^2 + 10 x+25.

EXAMPLE 4

Middle term of

Twice First term Last term

of binomial of binomial

2 #x# 5 = 10 x x 2 + 10 x+ 25

Since 10xis the middle term of the trinomial, the trinomial is a perfect square.

x^2 + 10 x+ 25 factors as 1 x+ 522. NOW TRY

Factoring Perfect Square Trinomials

Factor each trinomial.

(a)

The first and last terms are perfect squares or Check to see

whether the middle term of is twice the product of the first and last

terms of the binomial x- 11.

x^2 - 22 x+ 121

1121 = 112 1 - 11222.

x^2 - 22 x+ 121

EXAMPLE 5

Middle term of

Twice First Last

term term

2 #x# 1 - 112 = - 22 x x 2 - 22 x+ 121

Thus, is a perfect square trinomial.

factors as

Same sign

x^2 - 22 x+ 121 1 x- 1122.

x^2 - 22 x+ 121

NOW TRY ANSWER

4. 1 y+ 722

Notice that the sign of the second term in the squared binomial is the same as the

sign of the middle term in the trinomial.