The final result is in factored form because it is a product.
CHECK
FOIL (Section 5.5)✓ Rearrange terms to obtain
the original polynomial.= 2 x+ 6 +ax+ 3 a
= 2 x+ax+ 6 + 3 a
1 x+ 3212 + a 2
1 x+ 3212 + a 2
364 CHAPTER 6 Factoring and Applications
(b)
Group the terms.
Factor each group.Factor out.CHECK
FOIL✓ Rearrange terms to obtain the
original polynomial.= 6 ax+ 24 x+ a+ 4
= 6 ax+ a+ 24 x+ 4
1 a+ 4216 x+ 12
= 1 a+ 4216 x+ 12 a+ 4
= 6 x 1 a+ 42 + 11 a+ 42
= 16 ax+ 24 x 2 + 1 a+ 42
6 ax + 24 x+ a+ 4
(c)
Group the terms.
Factor each group.
Factor outCHECK
FOIL✓ Original polynomial
(d)
Group the terms.
Factor out so there is a common factor,Factor outCheckby multiplying. NOW TRY
= 1 t+ 221 t^2 32 t+2.
t+2;- 3 (t+2)=- 3 t-6.= t^2 1 t+ 22 3 1 t+ 22 3
= 1 t^3 + 2 t^22 + 1 - 3 t- 62
t^3 + 2 t^2 - 3 t- 6
= 2 x^2 - 10 x+ 3 xy- 15 y
= 2 x^2 + 3 xy- 10 x- 15 y
1 x- 5212 x+ 3 y 2
= 1 x- 5212 x+ 3 y 2 x-5.
= 2 x 1 x- 52 + 3 y 1 x- 52
= 12 x^2 - 10 x 2 + 13 xy- 15 y 2
2 x^2 - 10 x+ 3 xy- 15 y
Remember the 1.Write a sign
between the groups.+Be careful
with signs.CAUTION Be careful with signs when groupingin a problem like Example 5(d).
It is wise to check the factoring in the second step, as shown in the side comment in
that example, before continuing.
Factoring a Polynomial with Four Terms by Grouping
Step 1 Group terms.Collect the terms into two groups so that each group
has a common factor.
Step 2 Factor within groups.Factor out the greatest common factor from
each group.
Step 3 Factor the entire polynomial. Factor out a common binomial
factor from the results of Step 2.
Step 4 If necessary, rearrange terms. If Step 2 does not result in a
common binomial factor, try a different grouping.NOW TRY
EXERCISE 5
Factor by grouping.
(a)
(b)
(c)x^3 + 5 x^2 - 8 x- 40
12 xy+ 3 x+ 4 y+ 1ab+ 3 a+ 5 b+ 15NOW TRY ANSWERS
- (a)
(b)
(c) 1 x+ 521 x^2 - 82
14 y+ 1213 x+ 121 b+ 321 a+ 52http://www.ebook777.com
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