362 CHAPTER 6 Factoring and Applications
NOW TRY
EXERCISE 2
Find the greatest common
factor for each list of terms.
(a)
(b)m^3 n^5 , m^4 n^4 , m^5 n^2
25 k^3 , 15k^2 , 35k^5(b)
y^15 =y^15
x^3 y^7 =x^3 #y^7
x^7 y^5 =x^7 #y^5
x^4 y^2 =x^4 #y^2
x^4 y^2 , x^7 y^5 , x^3 y^7 , y^15
There is no xin the last term, so xwill not appear in
the greatest common factor. There is a yin each term,
however, and 2 is the least exponent on y.
GCF =y^2
y^15 ,
NOW TRYOBJECTIVE 2 Factor out the greatest common factor. Writing a polynomial
(a sum) in factored form as a product is called factoring.For example, the polynomial
has two terms: 3 mand 12. The greatest common factor of these two terms is 3. We
can write so that each term is a product with 3 as one factor.
Distributive propertyThe factored form of is This process is called factoring out the
greatest common factor.
3 m+ 12 31 m+ 42.
= 31 m+ 42
= 3 #m+ 3 # 4 GCF= 3
3 m+ 12
3 m+ 12
3 m+ 12
CAUTION The polynomial is notin factored form when written as
Not in factored formThe termsare factored, but the polynomial is not. The factored form of is
the product
31 m+ 42. In factored form
3 m+ 12
3 #m+ 3 #4.
3 m+ 12
Factoring Out the Greatest Common FactorWrite in factored form by factoring out the greatest common factor.
(a)
Distributive propertyCHECK Multiply the factored form.
Distributive property✓ Original polynomial
(b)
Factor out.CHECK
= 20 m^5 + 10 m^4 + 15 m^3 ✓ Original polynomial
5 m^314 m^2 + 2 m+ 32
= 5 m^314 m^2 + 2 m+ 32 5 m^3
= 5 m^314 m^22 + 5 m^312 m 2 + 5 m^3132 GCF= 5 m^3
20 m^5 + 10 m^4 + 15 m^3
= 5 y^2 + 10 y
= 5 y 1 y 2 + 5 y 122
5 y 1 y+ 22
= 5 y 1 y+ 22
= 5 y 1 y 2 + 5 y 122 GCF= 5 y
5 y^2 + 10 y
EXAMPLE 3
NOW TRY ANSWERS
- (a) 5 k^2 (b)m^3 n^2
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