8-7 Factor Completely
Name
Date
Factor completely: 3x3 ⫺ 24x2 ⫹ 48x
3x(x2 ⫺ 8x ⫹ 16)
3x(x ⫺ 4)2
Factor completely: 9v3 ⫺ 2
9v3 ⫺ 2
The GCF is 3x.
x2 ⫺ 8x ⫹ 16 is a
perfect-square trinomial.
?
Check: 3x3 ⫺ 24x2 ⫹ 48x ⫽ 3x(x ⫺ 4)2
?
⫽ 3x(x2 ⫺ 8x ⫹ 16)
⫽ 3x3 ⫺ 24x2 ⫹ 48x True
No GCF
Not the difference of
two perfect squares
9v3 ⫺ 2 cannot be factored.
It is prime.
Factor completely. Check using multiplication. If the polynomial cannot
be factored using integers, label it prime.
1. 2a2 ⫺ 162
2. 3b2 ⫺ 48
3. 8c3 ⫺ 80c2 ⫹ 200c
2(a2 ⴚ 81)
2(a ⴙ 9)(a ⴚ 9)
Check:
?
2a2 ⴚ 162 ⴝ 2(a ⴙ 9)(a ⴚ 9)
?
ⴝ (2a ⴙ 18)(a ⴚ 9)
ⴝ 2a2 ⴚ 162 True
4. 2f 3 ⫺ 28f 2 ⫹ 98f
2f(f 2 ⴚ 14f ⴙ 49)
Copyright © by William H. Sadlier, Inc. All rights reserved.
2f(f ⴚ
7)2
7. ⫺5x2y ⫺ 25xy ⫺ 15y
ⴚ5y(x2 ⴙ 5x ⴙ 3)
x2 ⴙ 5x ⴙ 3 is prime.
ⴚ5y(x2 ⴙ 5x ⴙ 3)
10. 4a3b ⫺ 9b2 ⫹ 4a
3(b2 ⴚ 16)
3(b ⴙ 4)(b ⴚ 4)
5. 3ab2 ⫹ 21ab ⫺ 54a
3a(b2 ⴙ 7b ⴚ 18)
3a(b ⴚ 2)(b ⴙ 9)
8. ⫺2hj2 ⫺ 18hj ⫺ 2h
8c(c ⴚ 5)2
6. 5vw2 ⫹ 20vw ⫺ 160v
5v(w2 ⴙ 4w ⴚ 32)
5v(w ⴚ 4)(w ⴙ 8)
9. 8x2y ⫺ 4x2 ⫹ 19y
ⴚ2h(j2 ⴙ 9j ⴙ 1)
j2 ⴙ 9j ⴙ 1 is prime.
ⴚ2h(j2 ⴙ 9j ⴙ 1)
11. 48t3 ⫹ 88t2 ⫹ 24t
8t(6t2 ⴙ 11t ⴙ 3)
ac ⴝ 18; 2 ⴙ 9 ⴝ 11
8t(6t2 ⴙ 2t ⴙ 9t ⴙ 3)
8t[2t(3t ⴙ 1) ⴙ 3(3t ⴙ 1)]
prime
8c(c2 – 10c ⴙ 25)
8t(2t ⴙ 3)(3t ⴙ 1)
Lesson 8-7, pages 216–217.
prime
12. 140m3 ⫹ 133m2 ⫹ 21m
7m(20m2 ⴙ 19m ⴙ 3)
ac ⴝ 60; 4 ⴙ 15 ⴝ 19
7m(20m2 ⴙ 4m ⴙ 15m ⴙ 3)
7m[4m(5m ⴙ 1) ⴙ 3(5m ⴙ 1)]
7m(5m ⴙ 1)(4m ⴙ 3)
Chapter 8
207
For More Practice Go To:
Factor completely. Check using multiplication. If the polynomial cannot
be factored using integers, label it prime.
14. 8c4 ⫺ 128
7(d4 ⴚ 1)
ⴙ 1)(d2 ⴚ 1)
7(d2
7(d2 ⴙ 1)(d ⴙ 1)(d ⴚ 1)
16. a4b ⫺ 5a2b ⫹ 4b
–
ⴙ 4)
d(a2 ⴚ 4)(a2 ⴚ 1)
b(a4
5a2
b(a ⴙ 2)(a ⴚ 2)(a ⴙ 1)(a ⴚ 1)
19. a4 ⫺ 2a2b2 ⫹ b4
(a2
(a2 ⴚ b2)2
ⴚ b2)(a2 ⴚ b2)
(a ⴙ b)2(a ⴚ b)2
15. c4d ⫺ 13c2d ⫹ 36d
d(c4 ⴚ 13c2 ⴙ 36)
d(c2 ⴚ 9)(c2 ⴚ 4)
8(c4 2 16)
1 4)(c2 2 4)
8(c2
8(c2 1 4)(c 1 2)(c 2 2)
d(c ⴙ 3)(c ⴚ 3)(c ⴙ 2)(c ⴚ 2)
17. 27a4 ⫹ 27a3 ⫺ 12a2 ⫺ 12a
18. 64x4 ⫹ 64x3 ⫺ 324x2 ⫺ 324x
3a(9a3 ⴙ 9a2 ⴚ 4a ⴚ 4)
3a[9a2(a ⴙ 1) ⴚ 4(a ⴙ 1)]
3a(9a2 ⴚ 4)(a ⴙ 1)
4x(16x3 ⴙ 16x2 ⴚ 81x ⴚ 81)
4x[16x2(x ⴙ 1) ⴚ 81(x ⴙ 1)]
4x(16x2 ⴚ 81)(x ⴙ 1)
3a(3a ⴚ 2)(3a ⴙ 2)(a ⴙ 1)
4x(4x ⴚ 9)(4x ⴙ 9)(x ⴙ 1)
20. 16x4 ⫺ 72x2y2 ⫹ 81y4
(4x2
(4x2 ⴚ 9y2)2
ⴚ 9y2)(4x2 ⴚ 9y2)
(2x ⴙ 3y)2(2x ⴚ 3y)2
21. x8 ⫺ 1
(x4 ⴚ 1)(x4 ⴙ 1)
(x2 ⴚ 1)(x2 ⴙ 1)(x4 ⴙ 1)
(x ⴚ 1)(x ⴙ 1)(x2 ⴙ 1)(x4 ⴙ 1)
22. A rectangular prism has a volume of
x3y ⫹ xy3 ⫺ 2x2y2 with a square base. If it
has a lateral area of 30 in.2, what is the
volume in cubic inches?
23. A rectangular prism has a volume of
x3 ⫺ y3 ⫹ x2y ⫺ xy2, and a square base.
If a lateral side has an area of 16 cm2,
what is the volume in cubic centimeters?
First factor: xy(x2 ⴚ 2xy ⴙ y2)
xy(x ⴚ y)2; the area of a lateral side is xy(x ⴚ y) ⴝ 30
5(2)(5 ⴚ 2) ⴝ 10(3) ⴝ 30; so x ⴝ 5, y ⴝ 2
V ⴝ 5(2)(3)2 ⴝ 90; The rectangular prism
has a volume of 90 in.3.
(x3 ⴙ x2y) ⴚ (xy2 ⴙ y3); x2(x ⴙ y) ⴚ y2(x ⴙ y)
(x2 ⴚ y2)(x ⴙ y); (x ⴙ y)(x ⴚ y)(x ⴙ y); (x ⴙ y)2(x ⴚ y); the
area of a lateral side is (x ⴙ y)(x ⴚ y) ⴝ 16 or x2 ⴚ y2 ⴝ 16
52 ⴚ 32 ⴝ 25 ⴚ 9 ⴝ 16; so x ⴝ 5, y ⴝ 3; V ⴝ (8)2(2) ⴝ 128
The rectangular prism has a volume of 128 cm3.
Multiply.
24. (24)(36) (24)(36) 5 (30 2 6)(30 1 6) 5 900 2 36 5 864
25. (91)(89) (91)(89) 5 (90 1 1)(90 2 1) 5 8100 2 1 5 8099
26. (74)(66) (74)(66) 5 (70 1 4)(70 2 4) 5 4900 2 16 5 4884
2
2
2
27. (29)2 (30 2 1) 5 (30) 2 2(30)(1) 1 (1) 5 900 2 60 1 1 5 840 1 1 5 841
2
2
2
28. (63)2 (60 1 3) 5 (60) 1 2(60)(3) 1 (3) 5 3600 1 360 1 9 5 3960 1 9 5 3969
208
Chapter 8
Copyright © by William H. Sadlier, Inc. All rights reserved.
13. 7d4 – 7