Factoring Out the Greatest Common FactorFactor out the greatest common factor.
(a)
The numerical part of the GCF is 3, the largest number that divides into both 9
and 12. The least exponent that appears on xis 2. The GCF is
Distributive property(b)
Distributive property(c)
= 3 k^411 - 5 k^3 + 8 k^52 GCF= 3 k^4
3 k^4 - 15 k^7 + 24 k^9
= 8 p^314 p- 3 + 5 p^22
= 8 p^314 p 2 + 8 p^31 - 32 + 8 p^315 p^22 GCF= 8 p^3
32 p^4 - 24 p^3 + 40 p^5
= 3 x 213 + 4 x 2
= 3 x 2132 + 3 x 214 x 2 GCF= 3 x 2
9 x^2 + 12 x 3
3 x^2.
9 x^2 + 12 x^3
EXAMPLE 2
6.1 Greatest Common Factors and Factoring by Grouping
NOW TRY
EXERCISE 2
Factor out the greatest
common factor.
20 m^3 n^3 - 15 m^3 n^2 + 10 m^2 nNOW TRY ANSWERS
- 5 m^2 n 14 mn 2 - 3 mn+ 22
Remember the 1.(d)
(e)
In each case, remember to check the factored form by multiplying. NOW TRY
Factoring Out a Binomial FactorFactor out the greatest common factor.
(a) The greatest common factor is
Factor out.Combine like terms.(b)
Factor out(c)
Factor out the common factor.= 1 r + 2 s 221 p- qr- 2 qs 2
= 1 r + 2 s 223 p-q 1 r + 2 s 24
p 1 r + 2 s 22 - q 1 r+ 2 s 23
= 1 m+ n 221 z^2 + x^221 m+n 22.
z^21 m+ n 22 +x^21 m+ n 22
= 1 x- 5213 x+ 112
= 1 x- 521 x+ 6 + 2 x+ 52
= 1 x- 5231 x+ 62 + 12 x+ 524 x- 5
1 x- 521 x+ 62 + 1 x- 5212 x+ 52 x-5.
EXAMPLE 3
= 5 y^315 x^2 + 6 y^2 - 3 x^4 y^42
25 x^2 y^3 + 30 y^5 - 15 x^4 y^7
= 6 m^2 n 14 mn - 3 +m^2 n^22
= 6 m^2 n 14 mn 2 + 6 m^2 n 1 - 32 + 6 m^2 n 1 m^2 n^22
24 m^3 n^2 - 18 m^2 n+ 6 m^4 n^3
Be careful with signs.(d)
Factor out
Distributive property
Combine like terms.
Look for a common factor.= - 21 p- 521 p+ 12 Commutative property NOW TRY
= 1 p- 523 - 21 p+ 124
= 1 p- 523 - 2 p- 24
= 1 p- 523 p+ 2 - 3 p- 44
= 1 p- 5231 p+ 22 - 13 p+ 424 p-5.
1 p- 521 p+ 22 - 1 p- 5213 p+ 42
NOW TRY
EXERCISE 3
Factor out the greatest
common factor.
(a)
(b)z 1 a-b 23 - 2 w 1 a-b 22
13 x- 2212 x- 5213 x- 221 x+ 12 +- (a)
(b) 1 a-b 221 za-zb- 2 w 2
13 x- 2213 x- 42