NOTE In the preceding example, we could have written 7xas rather than
as 3 x+ 4 x.Factoring by grouping would give the same answer. Try this.
4 x+ 3 x,
374 CHAPTER 6 Factoring and Applications
NOW TRY
EXERCISE 1
Factor.
(a)
(b)
(c) 8 x^2 - 2 xy- 3 y^2
15 m^2 +m- 22 z^2 + 5 z+ 3Factoring Trinomials by GroupingFactor each trinomial.
(a)
We must find two integers with a product of and a sum of 1.
Sum is 1.Product is.The integers are and 3. We write the middle term, r, as
Group the terms.The binomials must be the same.Factor outCHECK Multiply to obtain 13 r - 1212 r+ 12 6 r^2 + r-1. ✓
= 13 r- 12 12 r+ 12 3 r-1.
= 2 r 13 r- 12 + 113 r- 12
= 16 r^2 - 2 r 2 + 13 r- 12
= 6 r^2 - 2 r+ 3 r- 1 r=- 2 r+ 3 r
6 r^2 + r- 1
- 2 - 2 r+ 3 r.
61 - 12 =- 66 r^2 + 1 r- 1
61 - 12 =- 6
6 r^2 + r- 1
EXAMPLE 1
(b)
Look for two integers whose product is and whose sum is
The required integers are 3 and
Group the terms.Factor each group.Factor out.CHECK Multiply to obtain 14 z + 1213 z- 22 12 z^2 - 5 z -2. ✓
= 14 z+ 1213 z- 22 4 z+ 1
= 3 z 14 z+ 12 214 z + 12
= 112 z^2 + 3 z 2 + 1 - 8 z- 22
= 12 z^2 + 3 z - 8 z- 2 - 5 z= 3 z- 8 z
12 z^2 - 5 z- 2
- 8.
121 - 22 =- 24 - 5.
12 z^2 - 5 z- 2
Remember the 1.(c)
Two integers whose product is and whose sum is 1 are and 6.
Group the terms.Factor each group.Factor outCHECK Multiply to obtain ✓
NOW TRY12 m-n 215 m+ 3 n 2 10 m^2 + mn- 3 n^2.
= 12 m-n 215 m+ 3 n 2 2 m-n.
= 5 m 12 m- n 2 + 3 n 12 m-n 2
= 110 m^2 - 5 mn 2 + 16 mn - 3 n^22
= 10 m^2 - 5 mn+ 6 mn- 3 n^2 mn=- 5 mn+ 6 mn
10 m^2 +mn - 3 n^2
101 - 32 =- 30 - 5
10 m^2 + mn- 3 n^2
Be careful with signs.NOW TRY ANSWERS
- (a)
(b)
(c) 14 x- 3 y 212 x+y 2
13 m- 1215 m+ 2212 z+ 321 z+ 12http://www.ebook777.com
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