(b)
Difference of cubes
Factor.(c)
Difference of cubes
Factor.= 110 k- 3 n 21100 k^2 + 30 kn+ 9 n^22 Multiply. NOW TRY
= 110 k- 3 n 23110 k 22 + 110 k 213 n 2 + 13 n 224
= 110 k 23 - 13 n 23
1000 k^3 - 27 n^3
= 13 x- 2 y 219 x^2 + 6 xy+ 4 y^22
= 13 x- 2 y 2313 x 22 + 13 x 212 y 2 + 12 y 224
= 13 x 23 - 12 y 23
27 x^3 - 8 y^3
336 CHAPTER 6 Factoring
NOW TRY
EXERCISE 3
Factor each polynomial.
(a)
(b) 125 a^3 - 8 b^3
t^3 - 1NOW TRY ANSWERS
- (a)
(b) 15 a- 2 b 2125 a^2 + 10 ab+ 4 b^22
1 t- 121 t 2 +t+ 12OBJECTIVE 4 Factor a sum of cubes. While the binomial cannot be
factored with real numbers, a sum of cubes,such as x^3 +y^3 ,is factored as follows.
x^2 + y^2
Sum of Cubesx^3 y^3 1 xy 21 x^2 xyy^22
NOTEThe sign of the second term in the binomial factor of a sum or difference of
cubes is always the sameas the sign in the original polynomial. In the trinomial
factor, the first and last terms are always positive. The sign of the middle term is the
opposite ofthe sign of the second term in the binomial factor.
Factoring Sums of CubesFactor each polynomial.
(a)
Sum of cubes
Factor.(b)
Sum of cubes
Factor.
Multiply.(c)
Sum of cubes
Factor.
Multiply.(d)
Factor out the common factor.
Write as a sum of cubes.= 31 x+ 421 x^2 - 4 x+ 162 Factor.
= 31 x^3 + 432
= 31 x^3 + 642
3 x^2 + 192
= 15 t+ 6 s 22125 t 2 - 30 ts 2 + 36 s 42
= 15 t+ 6 s 22315 t 22 - 15 t 216 s 22 + 16 s 2224
= 15 t 23 + 16 s 223
125 t^3 + 216 s 6
= 13 z+ 5219 z^2 - 15 z+ 252
= 13 z+ 52313 z 22 - 13 z 2152 + 524
= 13 z 23 + 53
27 z^3 + 125
= 1 r+ 321 r^2 - 3 r+ 92 32 = 9
= 1 r+ 321 r^2 - 3 r+ 322
=r^3 + 33
r^3 + 27
EXAMPLE 4
Remember the
common factor.12 y 22 = 22 y^2 , not 2 y^2.13 x 22 = 32 x^2 , not 3 x^2.