8-3 Factor Trinomials ax 2 ⴙ bx ⴙ c with a fi 1
Name
Date
Factor: 5a2 ⫹ 16a ⫹ 3
Think
a ⫽ 5, b ⫽ 16, c ⫽ 3; ac ⫽ 15
Find two positive factors of 15
that sum to 16.
Factors of 15
1, 15
3, 5
5a2 ⫹ 16a ⫹ 3 ⫽ 5a2 ⫹ (15 ⫹ 1)a ⫹ 3
⫽ 5a2 ⫹ 15a ⫹ a ⫹ 3
⫽ (5a2 ⫹ 15a) ⫹ (a ⫹ 3)
⫽ 5a(a ⫹ 3) ⫹ 1(a ⫹ 3)
⫽ (5a ⫹ 1)(a ⫹ 3)
Sum of factors
16
8
Select the factors 15 and 1.
Use the Distributive Property.
Group terms that have a common monomial factor.
Factor each binomial using the greatest common
monomial factor.
Use the Distributive Property.
So in factored form, 5a2 ⫹ 16a ⫹ 3 ⫽ (5a ⫹ 1)(a ⫹ 3).
Multiply to check: (5a ⫹ 1)(a ⫹ 3) ⫽ 5a2 ⫹ 15a ⫹ a ⫹ 3
⫽ 5a2 ⫹ 16a ⫹ 3 ✓
Factor each trinomial. Check by multiplying the factors.
If the polynomial cannot be factored, write prime. Check students’ work.
1. 2x2 ⫹ 11x ⫹ 5
ac ⴝ 10; 1 ⴙ 10 ⴝ 11
2x2 ⴙ 10x ⴙ x ⴙ 5
2x(x ⴙ 5) ⴙ 1(x ⴙ 5)
Copyright © by William H. Sadlier, Inc. All rights reserved.
(2x ⴙ 1)(x ⴙ 5)
Check:
ⴝ 2x2 ⴙ 10x ⴙ x ⴙ 5
ⴝ 2x2 ⴙ 11x ⴙ 5
5. 3t2 ⫹ 14t ⫺ 8
2. 2r2 ⫹ 15r ⫹ 7
ac ⴝ 14; 1 ⴙ 14 ⴝ 15
2r2 ⴙ 14r ⴙ r ⴙ 7
2r(r ⴙ 7) ⴙ 1(r ⴙ 7)
6. 4q2 ⫹ 29q ⫺ 25
9. 3n2 ⫺ n ⫺ 10
ac ⴝ ⴚ30; ⴚ6 ⴙ 5 ⴝ ⴚ1
3n2 ⴚ 6n ⴙ 5n ⴚ 10
3n(n ⴚ 2) ⴙ 5(n ⴚ 2)
(3n ⴙ 5)(n ⴚ 2)
prime
10. 7x2 ⫺ 5x ⫺ 12
ac ⴝ ⴚ84; ⴚ12 ⴙ 7 ⴝ ⴚ5
7x2 ⴙ 7x ⴚ 12x ⴚ 12
7x(x ⴙ 1) ⴚ 12(x ⴙ 1)
(7x ⴚ 12)(x ⴙ 1)
4. 2p2 ⫺ 9p ⫹ 7
ac ⴝ 6; ⴚ2 ⴙ (ⴚ3) ⴝ ⴚ5
2c2 ⴚ 2c ⴚ 3c ⴙ 3
2c(c ⴚ 1) ⴚ 3(c ⴚ 1)
ac ⴝ 14; ⴚ2 ⴙ (ⴚ7) ⴝ ⴚ9
2p2 ⴚ 2p ⴚ 7p ⴙ 7
2p(p ⴚ 1) ⴚ 7(p ⴚ 1)
7. 5a2 ⫹ 13a ⫺ 6
8. 11x2 ⫹ 9x ⫺ 2
(2c ⴚ 3)(c ⴚ 1)
(2r ⴙ 1)(r ⴙ 7)
ac ⴝ ⴚ100; ⴚ1 ⴙ 100 ⴝ 99
ac ⴝ ⴚ24; ⴚ1 ⴙ 24 ⴝ 23
ⴚ2 ⴙ 12 ⴝ 10; ⴚ3 ⴙ 8 ⴝ 5 ⴚ2 ⴙ 50 ⴝ 48; ⴚ4 ⴙ 25 ⴝ 21
ⴚ5 ⴙ 20 ⴝ 15; ⴚ10 ⴙ 10 ⴝ 0
ⴚ4 ⴙ 6 ⴝ 2
prime
3. 2c2 ⫺ 5c ⫹ 3
ac ⴝ ⴚ30
15 ⴙ (ⴚ2) ⴝ 13
5a2 ⴙ 15a ⴚ 2a ⴚ 6
5a(a ⴙ 3) ⴚ 2(a ⴙ 3)
(5a ⴚ 2)(a ⴙ 3)
11. 19y2 ⫺ 138y ⫹ 35
ac ⴝ 665
ⴚ133 ⴙ (ⴚ5) ⴝ ⴚ138
19y2 ⴚ 133y ⴚ 5y ⴙ 35
19y(y ⴚ 7) ⴚ 5(y ⴚ 7)
(19y ⴚ 5)(y ⴚ 7)
Lesson 8-3, pages 206–209.
(2p ⴚ 7)(p ⴚ 1)
ac ⴝ ⴚ22; 11 ⴙ (ⴚ2) ⴝ 9
11x2 ⴙ 11x ⴚ 2x ⴚ 2
11x(x ⴙ 1) ⴚ 2(x ⴙ 1)
(11x ⴚ 2)(x ⴙ 1)
12. 19a2 ⫺ 82a ⫹ 24
ac ⴝ 456; ⴚ76 ⴙ (ⴚ6) ⴝ ⴚ82
19a2 ⴚ 76a ⴚ 6a ⴙ 24
19a(a ⴚ 4) ⴚ 6(a ⴚ 4)
(19a ⴚ 6)(a ⴚ 4)
Chapter 8
199
For More Practice Go To:
Factor each trinomial. Check by multiplying the factors. If the polynomial
cannot be factored, write prime. Check students’ work.
13. 5x2 ⫹ 12x ⫹ 7
Find the value of the linear term:
12x
(5x ⴙ 7)(x ⴙ 1)
36x
(5x ⴙ 1)(x ⴙ 7)
14. 3a2 ⫹ 34a ⫹ 11
(5x ⴙ 7)(x ⴙ 1)
(3a ⴙ 1)(a ⴙ 11)
(3a ⴙ 11)(a ⴙ 1)
Check: ⴝ 5x2 ⴙ 5x ⴙ 7x ⴙ 7
ⴝ 5x2 ⴙ 12x ⴙ 7
16. 13c2 ⫺ 70c ⫹ 25
(13c ⴚ 1)(c ⴚ 25)
(13c ⴚ 25)(c ⴚ 1)
(13c ⴚ 5)(c ⴚ 5)
(8x ⴚ 15)(x ⴙ 1)
(4x ⴚ 15)(2x ⴙ 1)
(4x ⴚ 1)(2x ⴙ 15)
(4x ⴙ 3)(2x ⴚ 5)
(4x ⴙ 5)(2x ⴚ 3)
(3a ⴙ 1)(a ⴙ 11)
ⴚ326g
ⴚ38c
ⴚ70g
ⴚ7x
ⴚ26x
ⴚ58x
ⴚ14x
ⴚ2x
(8n ⴚ 1)(n ⴙ 3)
(8n ⴚ 3)(n ⴙ 1)
(2n ⴚ 3)(4n ⴙ 1)
(2n ⴙ 3)(4n ⴚ 1)
22. 45y2 ⫹ 56y ⫺ 45
20. 12z2 ⫺ 16z ⫺ 16
(12z ⴚ 16)(z ⴙ 1)
(2z ⴚ 1)(6z ⴙ 16)
(2z ⴚ 2)(6z ⴙ 8)
(4z ⴚ 2)(3z ⴙ 8)
(4z ⴚ 8)(3z ⴙ 2)
(9y ⴚ 5)(5y ⴙ 9)
(11g ⴚ 3)(g ⴚ 3)
(12b ⴚ 1)(b ⴙ 7)
(12b ⴚ 7)(b ⴙ 1)
(4b ⴚ 1)(3b ⴙ 7)
(4b ⴙ 7)(3b ⴚ 1)
23n
5n
–10n
10n
ⴚ4z
26z
4z
26z
ⴚ16z
(4z ⴚ 8)(3z ⴙ 2)
(15t ⴚ 9)(t ⴙ 8)
(15t ⴙ 12)(t ⴚ 6)
(5t ⴙ 9)(3t ⴚ 8)
(5t ⴚ 9)(3t ⴙ 8)
ⴚ100g
ⴚ20g
ⴚ36g
18. 12b2 ⫹ 17b ⫺ 7
23. 15t2 ⫹ 13t ⫺ 72
ⴚ120y
0y
ⴚ56y
56y
(11g ⴚ 1)(g ⴚ 9)
(11g ⴚ 9)(g ⴚ 1)
(11g ⴚ 3)(g ⴚ 3)
(2n ⴙ 3)(4n ⴚ 1)
(4x ⴙ 5)(2x ⴚ 3)
(15y ⴙ 5)(3y ⴚ 9)
(15y ⴚ 15)(3y ⴙ 3)
(9y ⴙ 5)(5y ⴚ 9)
(9y ⴚ 5)(5y ⴙ 9)
34a
14a
17. 8n2 ⫹ 10n ⫺ 3
(13c ⴚ 5)(c ⴚ 5)
19. 8x2 ⫺ 2x ⫺ 15
15. 11g2 ⫺ 36g ⫹ 9
83b
5b
25b
17b
(4b ⴙ 7)(3b ⴚ 1)
21. 10m2 ⫺ 31m ⫹ 63
(10m ⴚ 63)(m ⴚ 1)
(10m ⴚ 9)(m ⴚ 7)
(5m ⴚ 7)(2m ⴚ 9)
(5m ⴚ 9)(2m ⴚ 7)
prime
ⴚ73m
ⴚ79m
ⴚ59m
ⴚ53m
24. 39s2 ⫹ 19s ⫹ 2
111t
ⴚ78t
ⴚ13t
13t
(5t ⴚ 9)(3t ⴙ 8)
(39s ⴙ 1)(s ⴙ 2)
(39s ⴙ 2)(s ⴙ 1)
(13s ⴙ 1)(3s ⴙ 2)
(13s ⴙ 2)(3s ⴙ 1)
79s
41s
29s
19s
(13s ⴙ 2)(3s ⴙ 1)
25.
8g2
6g
12g
9
8g2 ⴙ 12g ⴙ 6g ⴙ 9
8g2 ⴙ 18g ⴙ 9
ac ⴝ 72; 6 ⴙ 12 ⴝ 18
8g2 ⴙ 12g ⴙ 6g ⴙ 9
4g(2g ⴙ 3) ⴙ 3(2g ⴙ 3)
(4g ⴙ 3)(2g ⴙ 3)
26.
12b2
10b
18b
15
12b2 ⴙ 18b ⴙ 10b ⴙ 15
12b2 ⴙ 28b ⴙ 15
12b2 ⴙ 18b ⴙ 10b ⴙ 15
6b(2b ⴙ 3) ⴙ 5(2b ⴙ 3)
(6b ⴙ 5)(2b ⴙ 3)
28. Factor the trinomial 8y4 ⫹ 32y3 ⫹ 30y2. Explain your steps.
27.
11m2
77m
4m
28
11m2 ⴙ 4m ⴙ 77m ⴙ 28
11m2 ⴙ 81m ⴙ 28
11m2 ⴙ 77m ⴙ 4m ⴙ 28
11m(m ⴙ 7) ⴙ 4(m ⴙ 7)
(11m ⴙ 4)(m ⴙ 7)
2y2(2y ⴙ 5)(2y ⴙ 3); Possible response: Factor out the greatest common monomial factor of the
trinomial, then factor the trinomial. The GCF of 8, 32, and 30 is 2
The GCF of y4, y3, and y2 is y2. So the greatest common monomial factor of the trinomial is 2y2
so 8y4 ⴙ 32y3 ⴙ 30y2 ⴝ 2y2(4y2) ⴙ 2y2(16y) ⴙ 2y2(15) ⴝ 2y2(4y2 ⴙ 16y ⴙ 15)
a ⴝ 4, b ⴝ 16, c ⴝ 15; ac ⴝ 60, factors of 60 / sum: 1, 60 / 61; 2, 30 / 32; 3, 20 / 23; 4, 15 / 19
5, 12 / 17; 6, 10 / 16; so 2y2(4y2 ⴙ 16y ⴙ 15) ⴝ 2y2(4y2 ⴙ 6y ⴙ 10y ⴙ 15) ⴝ 2y2[2y(2y ⴙ 3) ⴙ 5(2y ⴙ 3)]
ⴝ 2y2(2y ⴙ 5)(2y ⴙ 3).
200
Chapter 8
Copyright © by William H. Sadlier, Inc. All rights reserved.
Write the polynomial shown by each area model, and then factor.