Rearranging Terms before Factoring by GroupingFactor by grouping.
(a)
Factoring out the common factor of 2 from the first two terms and the common
factor of xfrom the last two terms gives the following.
This does not lead to a common factor, so we try rearranging the terms.
Commutative property
Group the terms.
Factor each group.
Rewrite
Factor outCHECK
FOIL✓ Original polynomial
(b)
We need to rearrange these terms to get two groups that each have a common
factor. Trial and error suggests the following grouping.
Group the terms.
Factor each group;Factor outSince the quantities in parentheses in the second step must be the same, we factored
out - 4 rather than 4. Checkby multiplying. NOW TRY
= 12 x- 321 y- 42 2 x-3.
- 412 x- 32 =- 8 x+12.
=y 12 x- 32 412 x- 32
= 12 xy- 3 y 2 + 1 - 8 x+ 122
2 xy+ 12 - 3 y- 8 x
2 xy+ 12 - 3 y- 8 x
= 10 x^2 - 12 y+ 15 x- 8 xy
= 10 x^2 + 15 x- 8 xy- 12 y
15 x- 4 y 212 x+ 32
= 15 x- 4 y 212 x+ 32 5 x- 4 y.
= 2 x 15 x- 4 y 2 + 315 x- 4 y 2 - 4 y+ 5 x.
= 2 x 15 x- 4 y 2 + 31 - 4 y+ 5 x 2
= 110 x^2 - 8 xy 2 + 1 - 12 y+ 15 x 2
= 10 x^2 - 8 xy- 12 y+ 15 x
10 x^2 - 12 y+ 15 x- 8 xy
= 215 x^2 - 6 y 2 +x 115 - 8 y 2
10 x^2 - 12 y+ 15 x- 8 xy
10 x^2 - 12 y+ 15 x- 8 xy
EXAMPLE 6
SECTION 6.1 The Greatest Common Factor; Factoring by Grouping 365
NOW TRY
EXERCISE 6
Factor by grouping.
(a)
(b) 5 xy- 6 - 15 x+ 2 y
12 p^2 - 28 q- 16 pq+ 21 pWrite a sign
between the groups.+Be careful
with signs.Complete solution available
on the Video Resources on DVD
6.1 EXERCISES
Find the greatest common factor for each list of numbers. See Example 1.
1.40, 20, 4 2.50, 30, 5 3.18, 24, 36, 48
4.15, 30, 45, 75 5.6, 8, 9 6.20, 22, 23Find the greatest common factor for each list of terms. See Examples 1 and 2.- 16 y, 24 8. 18 w, 27
- , , 10. , ,
- , 12. ,
- 12 m^3 n^2 , , 18 m^5 n^436 m^8 n^3 14. 25 p^5 r^7 , , 50 30 p^7 r^8 p^5 r^3
x^4 y^3 xy^2 a^4 b^5 a^3 b30 x^340 x^650 x^760 z^470 z^890 z^9NOW TRY ANSWERS
- (a)
(b) 15 x+ 221 y- 32
13 p- 4 q 214 p+ 72