Beginning Algebra, 11th Edition

Notice the pattern of the terms in the factored form of

• (a binomial factor)(a trinomial factor)


• The binomial factor has the difference of the cube roots of the given terms.


• The terms in the trinomial factor are all positive.


• The terms in the binomial factor help to determine the trinomial factor.

x^3 - y^3 =

x^3 - y^3.

SECTION 6.4 Special Factoring Techniques 385

NOW TRY
EXERCISE 6
Factor each polynomial.

(a)

(b)

(c)

(d) 125 x^3 - 343 y^6

3 k^3 - 192

8 t^3 - 125

a^3 - 27

positive

First term product of second term

squared the terms squared

x^3 - y^3 = 1 x- y 21 x^2 + xy + y^22

+ +

CAUTION The polynomial is not equivalent to.

= 1 x- y 21 x^2 - 2 xy+y^22

= 1 x-y 21 x^2 + xy+ y^22 = 1 x- y 21 x- y 21 x-y 2

x^3 - y^31 x- y 23

x^3 - y^31 x- y 23

Factoring Differences of Cubes

Factor each polynomial.

(a)

Let and in the pattern for the difference of cubes.

Let

(b)

and

Let

= 12 p- 3214 p^2 + 6 p+ 92 Apply the exponents. Multiply.

= 12 p- 32312 p 22 + 12 p 23 + 324 x= 2 p, y=3.

= 12 p 23 - 33 8 p^3 = 12 p 23 27 = 33.

8 p^3 - 27

= 1 m- 521 m^2 + 5 m+ 252 52 = 25

m^3 - 125 =m^3 - 53 = 1 m- 521 m^2 + 5 m+ 522 x=m, y=5.

x^3 - y^3 = 1 x - y 21 x^2 + xy +y^22

x=m y= 5

m^3 - 125

EXAMPLE 6

(c)

Factor out the common factor, 4.

Factor the difference of cubes.

(d)

Write each term as a cube.

Factor the difference of cubes.

Apply the exponents. Multiply.

NOW TRY

= 15 t- 6 s^22125 t^2 + 30 ts^2 + 36 s^42

= 15 t- 6 s^22315 t 22 + 5 t 16 s^22 + 16 s^2224

= 15 t 23 - 16 s^223

125 t^3 - 216 s^6

= 41 m- 221 m^2 + 2 m+ 42

= 41 m^3 - 232 8 = 23

= 41 m^3 - 82

4 m^3 - 32

NOT 2p^2.

12 p 22 = 22 p^2 = 4 p^2 ,

NOW TRY ANSWERS

6. (a)
   (b)
   (c)
   (d)
   125 x^2 + 35 xy^2 + 49 y^42

15 x- 7 y^22 #

31 k- 421 k^2 + 4 k+ 162

12 t- 5214 t^2 + 10 t+ 252

1 a- 321 a^2 + 3 a+ 92