OBJECTIVE 4 Factor a sum of cubes. A sum of squares, such as
cannot be factored by using real numbers, but a sum of cubescan.
m^2 +25,
386 CHAPTER 6 Factoring and Applications
CAUTION A common error in factoring a difference of cubes, such as
is to try to factor This is usually
not possible.
x^3 - y^3 = 1 x- y 21 x^2 + xy+y^22 , x^2 +xy+y^2.
Factoring a Sum of Cubesx^3 y^3 1 xy 21 x^2 xyy^22
PositiveDifference of cubesSame Opposite
sign signPositiveSum of cubesSame Opposite
sign signx^3 + y^3 = 1 x+ y 21 x^2 - xy+y^22
x^3 - y^3 = 1 x- y 21 x^2 + xy+y^22
The only difference between
the patterns is the positive
and negative signs.Factoring Sums of CubesFactor each polynomial.
(a)
Factor the sum of cubes.
Apply the exponent.(b)
and
Factor the sum of cubes.= 12 m+ 5 n 214 m^2 - 10 mn + 25 n^22
= 12 m+ 5 n 2312 m 22 - 2 m 15 n 2 + 15 n 224
= 12 m 23 + 15 n 23 8 m^3 = 12 m 23 125 n^3 = 15 n 23.
8 m^3 + 125 n^3
= 1 k+ 321 k^2 - 3 k+ 92
= 1 k+ 321 k^2 - 3 k+ 322
= k^3 + 33 27 = 33
k^3 + 27
EXAMPLE 7
Compare the pattern for the sumof cubes with that for the differenceof cubes.
(c)
Factor the sum of cubes.NOW TRY= 110 a^2 + 3 b 21100 a^4 - 30 a^2 b+ 9 b^22110 a^222 = 1021 a^222 = 100 a^4
= 110 a^2 + 3 b 23110 a^222 - 110 a^2213 b 2 + 13 b 224
= 110 a^223 + 13 b 23
1000 a^6 + 27 b^3
Be careful:
12 m 22 = 22 m^2 and. 15 n 22 = 52 n^2NOW TRY
EXERCISE 7
Factor each polynomial.
(a)
(b) 27 a^3 + 8 b^3
x^3 + 125NOW TRY ANSWERS
- (a)
(b) 13 a+ 2 b 219 a^2 - 6 ab+ 4 b^22
1 x+ 521 x^2 - 5 x+ 252http://www.ebook777.com
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