7.7 Perfect Squares and Factoring

CORD Math

Mrs. Spitz

Fall 2006

Objective

• Identify and factor perfect square trinomials.

Assignment

• Pg. 283 #4-42 all

Introduction

• Numbers such as 1, 4, 9 and 16 are called perfect squares since they can be expressed as the square of an integer. Products of the form (a + b)2 and (a – b)2 are also called perfect squares, and the expansions of these products are called perfect square trinomials.

Perfect Square Trinomials

222

222

2)(

2)(

bababa

bababa

These patterns can be used to help you factor trinomials.

222 8)8)((2)8( yyy

64162 yy

Finding a Product

These patterns can be used to help you factor trinomials.

222 8)8)((26416 yyyy

2)8( y

Factoring

These patterns can be used to help you factor trinomials.

222 )5()5)(2(2)2()52( zzxxzx

22 25204 zxzx

Finding a Product

These patterns can be used to help you factor trinomials.

22 25204 zxzx

2

22

)52(

)5()5)(2(2)2(

zx

zzxx

Factoring

Can it be factored?

• To determine whether a trinomial can be factored by using these patterns, you must first decide if it is a perfect square trinomial. In other words, you must determine whether it can be written in the form

222

222

2)(

2)(

bababa

bababa

Example 1: Determine whether x2 + 22x +121 is a perfect square trinomial. If so, factor it.

a. Is the first term a perfect square?

To determine whether x2 + 22x + 121 is a perfect square trinomial, answer each question.

YES

YES

YES

b. Is the last term a perfect square?

c. Is the middle term 2(a)(b)?

x2 (x)2

121 (11)2

22x 2(x)(11)

So,

x2 + 22x + 121 is a perfect square trinomial. It can be factored as follow:

222 11)11)((212122 xxxx2)11( x

Example 2: Determine whether 16a2 + 81 – 72a is a perfect square trinomial. If so, factor it.

a. Is the first term a perfect square?

First arrange the terms of 16a2 + 81 + 72a, so the powers of a are in descending order.

YES

YES

YES

b. Is the last term a perfect square?

c. Is the middle term 2(a)(b)?

16a2 (4a)2

81 (9)2

72a 2(4a)(9)

So,

16a2 – 72a + 81 is a perfect square trinomial. It can be factored as follow:

222 9)9)(4(2)4(817216 aaaa2)94( a

Example 3: Determine whether 9p2 - 56p + 49 is a perfect square trinomial. If so, factor it.

a. Is the first term a perfect square?

Follow the steps.

YES

YES

NO

b. Is the last term a perfect square?

c. Is the middle term 2(a)(b)?

d. ?

9p2 (3p)2

49 (7)2

56p 2(3p)(7)

9p2 - 56p + 49 is NOT a perfect square trinomial.

So,

16a2 – 72a + 81 is a perfect square trinomial. It can be factored as follow:

222 9)9)(4(2)4(817216 aaaa2)94( a

Example 4: Is it possible for 9x2 + 12xy + 4y2 is a perfect square trinomial? If so, what is the measure of each side of the square?

a. Is the first term a perfect square?

Follow the steps.

YES

YES

NO

b. Is the last term a perfect square?

c. Is the middle term 2(a)(b)?

9x2 (3x)2

4y2 (2y)2

12xy 2(3x)(47)

So,

2222 )2()2)(3(2)3(4129 yyxxyxyx

2)23( yx

• Yes. 9x2 +12xy + 4y2 is a perfect square trinomial. Each side is (3x + 2y).