Chapter Chapter 6 6 Section Section 3 3 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Chapter Chapter 66Section Section 33
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
More on Factoring Trinomials
Factor trinomials by grouping when the coefficient of the squared term is not 1.Factor trinomials by using the FOIL method.
11
22
6.36.36.36.3
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Trinomials such as 2x2 + 7x + 6, in which the coefficient of the squared term is not 1, are factored with extensions of the methods from the previous sections. One such method uses factoring by grouping from Section 6.1.
More on Factoring Trinomials
Slide 6.3 - 3
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 11
Factor trinomials by groupingwhen the coefficient of the
squaredterm is not 1.
Slide 6.3 - 4
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Factor trinomials by grouping when the coefficient of the squared term is not 1.
Recall that a trinomial such as m2 + 3m + 2 is factored by finding two numbers whose product is 2 and whose sum is 3. To factor 2x2 + 7x + 6, we look for two integers whose product is 2 · 6 = 12 and whose sum is 7.
Slide 6.3 - 5
22 7 6x x
Sum
Product is 2 · 6 = 12
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Factor trinomials by grouping when the coefficient of the squared term is not 1. (cont’d)
Slide 6.3 - 6
By considering pairs of positive integers whose product is 12, we find the necessary integers to be 3 and 4. We use these integers to write the middle term, 7x, as 7x = 3x + 4x. The trinomial 2x2 + 7x + 6 becomes
2 22 6 2 67 3 4 .x xx x x
22 3 4 6x x x
2 3 2 32x x x
2 2 3x x
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 1
Factor.
23 3 1 1p p p
Factoring Trinomials by Grouping
Slide 6.3 - 7
23 4 1p p
212 16 3z z
2 28 6 5r rt t
3 1 1p p
Solution:
23 3 1 1p p p
3 1 1 1p p p
212 18 2 3z z z 212 18 2 3z z z 6 2 3 1(2 3)z z z 6 1 2 3z z
2 28 10 4 5r rt rt t 2 28 10 4 5r rt rt t
2 4 5r t r t 2 4 5 1 4 5r r t t r t
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 2
Factor 6p4 + 21p3 + 9p2.
2 23 2 6 1 3p p p p
Solution:
Factoring a Trinomial with a Common Factor by Grouping
Slide 6.3 - 8
2 23 2 7 3p p p
2 23 2 6 1 3p p p p
23 2 1 3p p p
23 2 3 1 3p p p p
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 22
Factor trinomials by using theFOIL method.
Slide 6.3 - 9
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
To factor 2x2 + 7x + 6, again using an alternate method explained in Section 6.2, we use the FOIL method in reverse. We want to write the equation 2x2 + 7x + 6 as the product of two binomials.
Factor trinomials by using the FOIL method.
Slide 6.3 - 10
22 7 6 x x
The product of the two first terms of the binomials is 2x2. The possible factors of 2x2 are 2x and x or −2x and −x. Since all terms of the trinomial are positive, we consider only positive factors. Thus, we have
22 7 6 2 . x x x x
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
The product of the two last terms, 6, can be factored as 1 · 6, 6 · 1, 3 · 2, or 3 · 2. Try each pair to find the pair that gives the correct middle term, 7x.
Factor trinomials by using the FOIL method. (cont’d)
Slide 6.3 - 11
2 6 1x x 6x2x8x
x12x13x
2 1 6x x
If the terms of the original polynomial have greatest common factor 1, then all of that polynomials binomial factors also have GCF 1.
Incorrect Incorrect
Now try the number 2 and 3 as factors of 6. Because of the common factor 2 in 2x + 2, (2x + 2)(x + 3) will not work, so we try (2x + 3)(x + 2).
2 3 2x x 3x4x7x
Correct
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 3
Solution:
Factoring a Trinomial with All Positive Terms by Using FOIL
Slide 6.3 - 12
Factor 6p2 + 19p + 10.
3 2 2 5p p 4 p
15 p19 p
6 10 1 1p p 10 p6 p
16 p 2 10 3 1p p
13p2 p
15p
6 2 1 5p p 2 p
30 p32 p
IncorrectIncorrect
Incorrect Correct
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 4
Solution:
Factoring a Trinomial with a Negative Middle Term by Using FOIL
Slide 6.3 - 13
Factor 10m2 – 23m + 12.
2 3 5 4m m 15m8m
23m
2 12 5 1m m 60m2m
62m
10 2 1 6m m 2m60m
62mIncorrect Incorrect
Correct
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 5
Factor 5p2 + 13p – 6.
Solution:
Factoring a Trinomial with a Negative Last Term by Using FOIL
Slide 6.3 - 14
5 2 3p p 2 p
15p13p
5 3 2p p 3p15p12 p
CorrectIncorrect
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Factor 6m2 + 11mn – 10n2.
EXAMPLE 6 Factoring a Trinomial with Two Variables
Slide 6.3 - 15
3 2 2 5m n m n 4mn
15mn11mn
Solution:
6 10 1m n m n 10mn6mn
4mnCorrectIncorrect
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
EXAMPLE 7 Factoring Trinomials with Common Factors
Slide 6.3 - 16
Factor.
Solution:
4 3 228 58 30x x x 3 2 224 32 6x x y xy
22 14 29 152 xx x
22 7 3 2 5x x x
2 212 16 32 x yx xy
2 6 2 3x x y x y
6x35x
29x
2xy18xy
16xy
Related Documents
