(b)
= 718 m+ 5 p 2
56 m+ 35 p
320 CHAPTER 6 Factoring
OBJECTIVES Writing a polynomial as the product of two or more simpler polynomials is called
factoringthe polynomial. For example, the product of 3xand is
and can be factored as the product
Multiplying
FactoringNotice that both multiplying and factoring use the distributive property, but in oppo-
site directions. Factoring “undoes,” or reverses, multiplying.
OBJECTIVE 1 Factor out the greatest common factor. The first step in fac-
toring a polynomial is to find the greatest common factorfor the terms of the poly-
nomial. The greatest common factor (GCF)is the largest term that is a factor of all
terms in the polynomial.
For example, the greatest common factor for is 4, since 4 is the largest
term that is a factor of (divides into) both 8xand 12.
Factor 4 from each term.
Distributive propertyAs a check, multiply 4 and The result should be Using the distrib-
utive property this way is called factoring out the greatest common factor.
2 x+3. 8 x+12.
= 412 x+ 32
= 412 x 2 + 4132
8 x+ 12
8 x+ 12
15 x^2 - 6 x= 3 x 15 x- 22
3 x 15 x- 22 = 15 x^2 - 6 x
15 x^2 - 6 x 3 x 15 x- 22.
5 x- 2 15 x^2 - 6 x,
Greatest Common Factors and Factoring by Grouping
6.1
1 Factor out the
greatest common
factor.
2 Factor by grouping.(c) There is no common
factor other than 1.2 y+ 5
(d)
Identity property= 1211 + 2 z 2 12 is the GCF.
= 12 # 1 + 12 # 2 z
12 + 24 z
Remember to
write the 1.CHECK
Distributive property= 12 + 24 z ✓ Original polynomial NOW TRY
= 12112 + 1212 z 2
1211 + 2 z 2
NOW TRY
EXERCISE 1
Factor out the greatest
common factor.
(a)
(b) 2 k- 7
54 m- 45NOW TRY ANSWERS
- (a)
(b)There is no common factor
other than 1.
916 m- 52Factoring Out the Greatest Common FactorFactor out the greatest common factor.
(a)
Since 9 is the GCF, factor 9 from each term.
Distributive propertyCHECK Multiply to obtain 91 z- 22 9 z- 18. Original polynomial
= 91 z- 22
= 9 #z- 9 # 2 GCF= 9