Beginning Algebra, 11th Edition

382 CHAPTER 6 Factoring and Applications

Factoring Differences of Squares

Factor each difference of squares.

(a)

(b)

Write each term as a square.

= 17 z+ 8 t 217 z- 8 t 2 Factor the difference of squares. NOW TRY

= 17 z 22 - 18 t 22

49 z^2 - 64 t^2

25 m^2 - 16 = 15 m 22 - 42 = 15 m+ 4215 m- 42

x 2 - y 2 = 1 x + y 21 x - y 2

NOW TRY EXAMPLE 2

EXERCISE 2
Factor each difference of
squares.

(a)

(b) 36 a^2 - 49 b^2

9 t^2 - 100

NOW TRY
EXERCISE 3
Factor completely.

(a)

(b)

(c)v^4 - 625

m^4 - 144

16 k^2 - 64

NOTE Always check a factored form by multiplying.

Factoring More Complex Differences of Squares

Factor completely.

(a)

Factor out the GCF, 9.

Write each term as a square.

Factor the difference of squares.

(b)

Write each term as a square.

= 1 p^2 + 621 p^2 - 62 Factor the difference of squares.

= 1 p^222 - 62

p^4 - 36

= 913 y+ 2213 y- 22

= 9313 y 22 - 224

= 919 y^2 - 42

81 y^2 - 36

EXAMPLE 3

Neither binomial can

be factored further.

(c)

Factor the difference of squares.

Factor the difference of squares again.

NOW TRY

= 1 m^2 + 421 m+ 221 m- 22

= 1 m^2 + 421 m^2 - 42

= 1 m^222 - 42

m^4 - 16

Don’t stop

here.

CAUTION Factor again when any of the factors is a difference of squares,

as in Example 3(c).Check by multiplying.

OBJECTIVE 2 Factor a perfect square trinomial.The expressions 144,

and are called perfect squaresbecause

and

A perfect square trinomialis a trinomial that is the square of a binomial. For

example, is a perfect square trinomial because it is the square of the

binomial

= 1 x+ 422

= 1 x+ 421 x+ 42

x^2 + 8 x+ 16

x+4.

x^2 + 8 x+ 16

144 = 122 , 4 x^2 = 12 x 22 , 81 m^6 = 19 m^322.

81 m^6

4 x^2 ,

NOW TRY ANSWERS

2. (a)
   (b)


3. (a)
   (b)
   (c) 1 v^2 + 2521 v+ 521 v- 52

1 m^2 + 1221 m^2 - 122

161 k+ 221 k- 22

16 a+ 7 b 216 a- 7 b 2

13 t+ 10213 t- 102

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