(c)
Checkmentally by distributing x^3 over each term inside the parentheses.
= x 3 1 x^2 + 12
= x 3 1 x^22 + x 3 112 GCF=x^3
x^5 + x^3
SECTION 6.1 The Greatest Common Factor; Factoring by Grouping 363
NOW TRY
EXERCISE 3
Write in factored form by
factoring out the greatest
common factor.
(a)
(b)
(c) 30 m^4 n^3 - 42 m^2 n^2
8 x^6 - 20 x^5 + 28 x^47 t^4 - 14 t^3Don’t forget the 1.(d)
= 4 m^3 p^215 m^4 - 9 p^22 Factor out 4 m^3 p^2. NOW TRY
= 4 m^3 p^215 m^42 - 4 m^3 p^219 p^22 GCF= 4 m^3 p^2
20 m^7 p^2 - 36 m^3 p^4
CAUTION Be sure to include the 1 in a problem like Example 3(c).Check that
the factored form can be multiplied out to give the original polynomial.
NOW TRY
EXERCISE 4
Write in factored form by
factoring out the greatest
common factor.
(a)
(b)a 1 t+ 102 - b 1 t+ 102
x 1 x+ 22 + 51 x+ 22Factoring Out the Greatest Common FactorWrite in factored form by factoring out the greatest common factor.
Same(a) The binomial is the greatest common factor.
Factor out.(b)
= 1 x+ 121 x^2 - 52 Factor out x+ 1. NOW TRY
x^21 x+ 12 - 51 x+ 12
= 1 a+ 321 a+ 42 a+ 3
a 1 a+ 32 + 41 a+ 32 a+ 3
EXAMPLE 4
NOTE In factored forms like those in Example 4,the order of the factors does not
matter because of the commutative property of multiplication.
1 a+ 321 a+ 42 can also be written 1 a+ 421 a+ 32.
OBJECTIVE 3 Factor by grouping.When a polynomial has four terms, com-
mon factors can sometimes be used to factor by grouping.
Factoring by GroupingFactor by grouping.
(a)
Group the first two terms and the last two terms, since the first two terms have a
common factor of 2 and the last two terms have a common factor of a.
Group the terms.
Factor each group.The expression is still not in factored form because it is the sumof two terms. Now,
however, is a common factor and can be factored out.
is a common factor.= 1 x+ 3212 + a 2 Factor out x+ 3.
= 21 x+ 32 +a 1 x+ 32 x+ 3
x+ 3
= 21 x+ 32 +a 1 x+ 32
= 12 x+ 62 + 1 ax+ 3 a 2
2 x+ 6 +ax + 3 a
2 x+ 6 + ax+ 3 a
EXAMPLE 5
is also correct.12 +a 21 x+ 32NOW TRY ANSWERS
- (a)
(b)
(c) - (a)
(b) 1 t+ 1021 a-b 2
1 x+ 221 x+ 526 m^2 n^215 m^2 n- 724 x^412 x^2 - 5 x+ 727 t^31 t- 22