340 CHAPTER 6 Factoring
Factoring Out a Common FactorFactor each polynomial.
EXAMPLE 1
(b)
= 4 mp 12 mp + 12
(a) 8 m^2 p^2 + 4 mp
= 91 p+ 52 GCF= 9
9 p+ 45
(c)
= 1 a+ b 215 x- y 2 Factor out a+b. NOW TRY
5 x 1 a+b 2 - y 1 a+b 2
OBJECTIVE 2 Factor binomials. For binomials, use one of the following rules.
Factoring a BinomialFor a binomial(two terms), check for the following patterns.
Difference of squares
Difference of cubes
Sum of cubes x^3 y^3 1 xy 21 x^2 xyy^22
x^3 y^3 1 xy 21 x^2 xyy^22
x^2 y^2 1 xy 21 xy 2
NOW TRY
EXERCISE 1
Factor each polynomial.
(a)
(b) 8 y 1 m-n 2 - 51 m-n 2
21 x^3 y^2 - 27 x^2 y^4NOW TRY ANSWERS
- (a)
(b) 1 m-n 218 y- 52
3 x^2 y^217 x- 9 y^22Factoring BinomialsFactor each binomial if possible.
(a)
Difference of squares(b)
Difference of cubes(c)
Sum of cubes(d) 25 m^2 + 121 is prime. It is a sumof squares. NOW TRY
= 110 m+ 121100 m^2 - 10 m+ 12 110 m 22 = 10 2 m^2 = 100 m^2
= 110 m+ 123110 m 22 - 110 m 2112 + 124 x^3 +y^3 = 1 x+y 21 x^2 - xy+y^22
= 110 m 23 + 13
1000 m^3 + 1
= 12 p- 3214 p^2 + 6 p+ 92 12 p 22 = 22 p^2 = 4 p^2
= 12 p- 32312 p 22 + 12 p 2132 + 324 x 3 - y^3 = 1 x-y 21 x^2 +xy+y^22
= 12 p 23 - 33
8 p^3 - 27
= 18 m+ 3 n 218 m- 3 n 2 x^2 - y^2 = 1 x+y 21 x-y 2
= 18 m 22 - 13 n 22
64 m^2 - 9 n^2
NOW TRY EXAMPLE 2
EXERCISE 2
Factor each binomial if
possible.
(a)
(b)
(c) 27 v^3 - 1000
9 x^2 + 1004 a^2 - 49 b^2- (a)
(b)prime
(c) 13 v- 10219 v^2 + 30 v+ 1002
12 a+ 7 b 212 a- 7 b 2OBJECTIVE 1 Factor out any common factor. This step is always the same,
regardless of the number of terms in the polynomial.
NOTE The binomial is a sum of squares. It can be factored, however,
because the greatest common factor of the terms is not 1.
Factor out the common factor 25.
This sum of squares cannot
be factored further.