Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson

Chapter 6

Factoring and Quadratic Equations - all with Video Answers

Educators

Section 1

The Greatest Common Factor; Factoring by Grouping 432

Problem 1

Fill in the blanks.
To ___________ a polynomial means to express it as a product of two (or more) polynomials.

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Problem 2

GCF stands for_______________. When we write $2 x+4$ as $2(x+2),$ we say that we have ______________ out the $\mathrm{GCF}, 2$.

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Problem 3

Fill in the blanks.
To factor $m^{3}+3 m^{2}+4 m+12$ by __________ we begin by writing $m^{2}(m+3)+4(m+3)$

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Problem 4

Fill in the blanks.
The terms $x(x-1)$ and $4(x-1)$ have the common __________ factor $x-1$

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Problem 5

Complete each factorization.
$$
\text { a. } 6 x=2 \cdot \quad \cdot x
$$
$$
\text { b. } 35 h^{2}=5 \cdot \quad \cdot h
$$
$$
\text { c. } 18 y^{3} z=2 \cdot \quad \cdot 3 \cdot \quad \cdot y \cdot \quad \cdot z
$$

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Problem 6

a. Find the GCF of $30 x^{2}$ and $105 x^{3}$.
$$
\begin{array}{l}
{30 x^{2}=2 \cdot 3 \cdot 5 \cdot x \cdot x} \\
{105 x^{3}=3 \cdot 5 \cdot 7 \cdot x \cdot x \cdot x}
\end{array}
$$
b. Find the GCF of $12 a^{2} b^{2}, 15 a^{3} b,$ and $75 a^{4} b^{2}$
$$
\begin{array}{l}
{12 a^{2} b^{2}=2 \cdot 2 \cdot 3 \cdot a \cdot a \cdot b \cdot b} \\
{15 a^{3} b=3 \cdot 5 \cdot a \cdot a \cdot a \cdot b} \\
{75 a^{4} b^{2}=3 \cdot 5 \cdot 5 \cdot a \cdot a \cdot a \cdot a \cdot b \cdot b}
\end{array}
$$

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Problem 7

a. Write a binomial such that the GCF of its terms is 2 .
b. Write a trinomial such that the GCF of its terms is $x$

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Problem 8

Check to determine whether each factorization is correct.
a. $9 y^{3}+5 y^{2}-15 y=3 y\left(3 y^{2}+2 y-5\right)$
b. $3 s^{3}+2 s^{2}+6 s+4=(3 s+2)\left(s^{2}+2\right)$

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Problem 9

Fill in the blanks to complete each factorization.
$$
\begin{aligned}
2 x^{2}+6 x &=2 x \cdot x+2 x \cdot 3 \\
&= ( )
\end{aligned}
$$

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Problem 10

Fill in the blanks to complete each factorization.
$3 t^{3}-t^{2}+15 t-5=t^{2}(3 t-1)+5(3 t-1)$
= ( ) ( )

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Problem 11

Consider the polynomial $2 k-8+h k-4 h$
a. How many terms does the polynomial have?
b. Is there a common factor of all the terms, other than $1 ?$
c. What is the GCF of the first two terms and what is the GCF
of the last two terms?

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Problem 12

What is the first step in factoring $8 y^{2}-16 y z-6 y+12 z ?$

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Problem 13

Complete each factorization.
$$
8 m^{2}-32 m+16=\quad\left(m^{2}-4 m+2\right)
$$

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Problem 14

Complete each factorization.
$$
10 a^{4}-15 a^{3}=5 a \quad(2 a-3)
$$

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Problem 15

Complete each factorization.
$$
\begin{aligned}
b^{3}-6 b^{2}+2 b-12 &=\quad(b-6)+\quad(b-6) \\
&=(\quad)\left(b^{2}+2\right)
\end{aligned}
$$

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Problem 16

Complete each factorization.
$$
\begin{aligned}
12+8 n-3 m-2 m n &=4(3+2 n) \quad m(3+2 n) \\
&=(3+2 n)(4 \quad m)
\end{aligned}
$$

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Problem 17

Find the GCF of each list of numbers.
$$
6,10
$$

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Problem 18

Find the GCF of each list of numbers.
$$
10,15
$$

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Problem 19

Find the GCF of each list of numbers.
$$
18,24
$$

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Problem 20

Find the GCF of each list of numbers.
$$
60,72
$$

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Problem 21

Find the GCF of each list of numbers.
$$
14,21,42
$$

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Problem 22

Find the GCF of each list of numbers.
$$
16,24,48
$$

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Problem 23

Find the GCF of each list of numbers.
$$
40,32,24
$$

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Problem 24

Find the GCF of each list of numbers.
$$
28,35,21
$$

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Problem 25

Find the GCF of each list of terms.
$$
m^{4}, m^{3}
$$

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Problem 26

Find the GCF of each list of terms.
$$
c^{2}, c^{7}
$$

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Problem 27

Find the GCF of each list of terms.
$$
15 x, 25
$$

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Problem 28

Find the GCF of each list of terms.
$$
9 a, 21
$$

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Problem 29

Find the GCF of each list of terms.
$$
20 c^{2}, 12 c
$$

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Problem 30

Find the GCF of each list of terms.
$$
18 r, 27 r^{3}
$$

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Problem 31

Find the GCF of each list of terms.
$$
18 a^{4}, 9 a^{3}, 27 a^{3}
$$

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Problem 32

Find the GCF of each list of terms.
$$
33 m^{5}, 22 m^{6}, 11 m^{5}
$$

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Problem 33

Find the GCF of each list of terms.
$$
24 a^{2}, 16 a^{3} b, 40 a b
$$

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Problem 34

Find the GCF of each list of terms.
$$
12 r^{2}, 15 r s, 9 r^{2} s^{2}
$$

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Problem 35

Find the GCF of each list of terms.
$$
6 m^{4} n, 12 m^{3} n^{2}, 9 m^{3} n^{3}
$$

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Problem 36

Find the GCF of each list of terms.
$$
15 c^{2} d^{4}, 10 c^{2} d, 40 c^{3} d^{3}
$$

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Problem 37

Find the GCF of each list of terms.
$$
4(x+7), 9(x+7)
$$

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Problem 38

Find the GCF of each list of terms.
$$
2(y-1), 5(y-1)
$$

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Problem 39

Find the GCF of each list of terms.
$$
4(p-t), p(p-t)
$$

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Problem 40

Find the GCF of each list of terms.
$$
a(b+c), 3(b+c)
$$

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Problem 41

Factor out the GCF.
$$
3 x+6
$$

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Problem 42

Factor out the GCF.
$$
18 x+24
$$

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Problem 43

Factor out the GCF.
$$
18 m-9
$$

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Problem 44

Factor out the GCF.
$$
15 s-35
$$

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Problem 45

Factor out the GCF.
$$
d^{2}-7 d
$$

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Problem 46

Factor out the GCF.
$$
a^{2}+9 a
$$

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Problem 47

Factor out the GCF.
$$
15 c^{3}+25
$$

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Problem 48

Factor out the GCF.
$$
33 h^{4}-22
$$

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Problem 49

Factor out the GCF.
$$
24 a-16 a^{2}
$$

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Problem 50

Factor out the GCF.
$$
18 r-30 r^{2}
$$

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Problem 51

Factor out the GCF.
$$
14 x^{2}-7 x-7
$$

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Problem 52

Factor out the GCF.
$$
27 a^{2}-9 a+9
$$

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Problem 53

Factor out the GCF.
$$
t^{4}+t^{3}+2 t^{2}
$$

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Problem 54

Factor out the GCF.
$$
b^{4}-b^{3}-3 b^{2}
$$

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Problem 55

Factor out the GCF.
$$
21 x^{2} y^{3}+3 x y^{2}
$$

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Problem 56

Factor out the GCF.
$$
3 x^{2} y^{3}-9 x^{4} y^{3}
$$

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Problem 57

Factor out $-1$ from each polynomial.
$$
-a-b
$$

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Problem 58

Factor out $-1$ from each polynomial.
$$
-x-2 y
$$

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Problem 59

Factor out $-1$ from each polynomial.
$$
-x^{2}-x+16
$$

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Problem 60

Factor out $-1$ from each polynomial.
$$
-t^{2}-9 t+1
$$

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Problem 61

Factor out $-1$ from each polynomial.
$$
5-x
$$

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Problem 62

Factor out $-1$ from each polynomial.
$$
10-m
$$

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Problem 63

Factor out $-1$ from each polynomial.
$$
9-4 a
$$

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Problem 64

Factor out $-1$ from each polynomial.
$$
7-8 b
$$

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Problem 65

Factor each polynomial by factoring out the opposite of the GCF.
$$
-3 x^{2}-6 x
$$

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Problem 66

Factor each polynomial by factoring out the opposite of the GCF.
$$
-4 a^{2}-6 a
$$

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Problem 67

Factor each polynomial by factoring out the opposite of the GCF.
$$
-4 a^{2} b+12 a^{3}
$$

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Problem 68

Factor each polynomial by factoring out the opposite of the GCF.
$$
-25 x^{4}+30 x^{2}
$$

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Problem 69

Factor each polynomial by factoring out the opposite of the GCF.
$$
-24 x^{4}-48 x^{3}+36 x^{2}
$$

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Problem 70

Factor each polynomial by factoring out the opposite of the GCF.
$$
-28 a^{5}-42 a^{4}+14 a^{3}
$$

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Problem 71

Factor each polynomial by factoring out the opposite of the GCF.
$$
-4 a^{3} b^{2}+14 a^{2} b^{2}-10 a b^{2}
$$

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Problem 72

Factor each polynomial by factoring out the opposite of the GCF.
$$
-30 x^{4} y^{3}+24 x^{3} y^{2}-60 x^{2} y
$$

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Problem 73

Factor.
$$
y(x+2)+3(x+2)
$$

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Problem 74

Factor.
$$
r(t+v)+3(t+v)
$$

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Problem 75

Factor.
$$
m(p-q)-5(p-q)
$$

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Problem 76

Factor.
$$
a b(c-7)-12(c-7)
$$

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Problem 77

Factor by grouping.
$$
2 x+2 y+a x+a y
$$

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Problem 78

Factor by grouping.
$$
b x+b z+5 x+5 z
$$

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Problem 79

Factor by grouping.
$$
r s-r u+8 s w-8 u w
$$

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Problem 80

Factor by grouping.
$$
12 a b-4 a c+3 d b-d c
$$

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Problem 81

Factor by grouping.
$$
7 m^{3}-2 m^{2}+14 m-4
$$

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Problem 82

Factor by grouping.
$$
9 s^{3}-2 s^{2}+36 s-8
$$

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Problem 83

Factor by grouping.
$$
5 x^{3}-x^{2}+10 x-2
$$

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Problem 84

Factor by grouping.
$$
6 a^{3}-a^{2}+18 a-3
$$

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Problem 85

Factor by grouping.
$$
a b+a c+b+c
$$

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Problem 86

Factor by grouping.
$$
x y+3 y^{2}+x+3 y
$$

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Problem 87

Factor by grouping.
$$
r s+4 s^{2}-r-4 s
$$

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Problem 88

Factor by grouping.
$$
t x+t z-x-z
$$

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Problem 89

Factor by grouping.
$$
2 a x+2 b x-3 a-3 b
$$

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Problem 90

Factor by grouping.
$$
r x+s x-r y-s y
$$

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Problem 91

Factor by grouping.
$$
m p-n p-m q+n q
$$

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Problem 92

Factor by grouping.
$$
9 p-9 q-m p+m q
$$

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Problem 93

Factor by grouping.
$$
5 m^{3}+6+5 m^{2}+6 m
$$

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Problem 94

Factor by grouping.
$$
4 t^{3}+14+28 t^{2}+2 t
$$

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Problem 95

Factor by grouping.
$$
y^{3}-12+3 y-4 y^{2}
$$

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Problem 96

Factor by grouping.
$$
h^{3}-8+h-8 h^{2}
$$

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Problem 97

Factor by grouping. Remember to factor out the GCF first.
$$
a x^{3}-2 a x^{2}+5 a x-10 a
$$

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Problem 98

Factor by grouping. Remember to factor out the GCF first.
$$
x^{3} y^{2}-2 x^{2} y^{2}+3 x y^{2}-6 y^{2}
$$

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Problem 99

Factor by grouping. Remember to factor out the GCF first.
$$
6 x^{3}-6 x^{2}+12 x-12
$$

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Problem 100

Factor by grouping. Remember to factor out the GCF first.
$$
3 x^{3}-6 x^{2}+15 x-30
$$

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Problem 101

Factor.
$$
h^{2}(14+r)+5(14+r)
$$

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Problem 102

Factor.
$$
x(y+9)-21(y+9)
$$

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Problem 103

Factor.
$$
22 a^{3}-33 a^{2}
$$

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Problem 104

Factor.
$$
39 r^{3}+26 r^{2}
$$

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Problem 105

Factor.
$$
a x+b x-a-b
$$

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Problem 106

Factor.
$$
2 x y+y^{2}-2 x-y
$$

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Problem 107

Factor.
$$
15 r^{8}-18 r^{6}-30 r^{5}
$$

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Problem 108

Factor.
$$
24 c m-12 c n+16 c
$$

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Problem 109

Factor.
$$
27 m p+9 m q-9 n p-3 n q
$$

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Problem 110

Factor.
$$
4 a b c+4 a c^{2}-2 b c-2 c^{2}
$$

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Problem 111

Factor.
$$
-60 p^{2} t^{2}-80 p t^{3}
$$

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Problem 112

Factor.
$$
-25 x^{5} y^{7}+75 x^{3} y^{2}
$$

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Problem 113

Factor.
$$
-2 x+5
$$

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Problem 114

Factor.
$$
-3 x+8
$$

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Problem 115

Factor.
$$
6 x^{2}-2 x y-15 x+5 y
$$

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Problem 116

Factor.
$$
m^{3}+5 m^{2}+m+5
$$

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Problem 117

Factor.
$$
2 x^{3} z-4 x^{2} z+32 x z-64 z
$$

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Problem 118

Factor.
$$
4 a^{2} b+12 a^{2}-8 a b-24 a
$$

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Problem 119

Factor.
$$
12 u v w^{3}-54 u v^{2} w^{2}
$$

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Problem 120

Factor.
$$
14 x y z-16 x^{2} y^{2} z
$$

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Problem 121

Factor.
$$
x^{3}+x^{2}+x+1
$$

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Problem 122

Factor.
$$
m^{4}+m^{3}+2 m+2
$$

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Problem 123

Factor.
$$
-3 r+2 s-3
$$

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Problem 124

Factor.
$$
-6 y z+12 x z+5 x y
$$

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Problem 125

Factor.
$$
\text { a. } 5 t^{3}+6 t^{2}+15 t+18
$$
$$
\text { b. } 3 t^{3}+6 t^{2}+15 t+18
$$

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Problem 126

Factor.
$$
\text { a, } x^{2}+x y+x+y
$$
$$
\text { b. } 2 x^{2}+2 x y+2 x+2 y
$$

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Problem 127

The dimensions of the rectangle shown below can be found by factoring the polynomial that represents its area. Find the polynomials that represent the length and the width of the rectangle.
$$Area =\left(x^{3}+4 x^{2}+5 x+20\right) \pi^{2}$$

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Problem 128

Find a polynomial that represents the amount of surface on the front face of the door shown on the right that needs to be stained. Then factor the polynomial.
(FIGURE CAN'T COPY)

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Problem 129

A teacher is going to purchase a block of wax so that the children in her classroom can melt it down, add coloring, pour it into a mold, and make their own crayons. The amount of wax to make one crayon is given by the volume formula:
$$
V=\pi r^{2} h_{1}+\frac{1}{3} \pi r^{2} h_{2}
$$
a. Rewrite the formula by factoring the expression on the right side.
b. Use the formula to estimate what size block of wax (in cubic inches) she needs to purchase if there are 20 students in her class, and each student will get 5 crayons. The dimensions for the crayon molds they will use are: $h_{1}=1 \frac{5}{6}$ in. $h_{2}=\frac{1}{2} \mathrm{in.},$ and $r=\frac{1}{4}$ in. Round up to the nearest cubic inch.

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Problem 130

The expression $\pi r s+\pi R s$ can be used to find the amount of material needed to make the lamp shade shown. Factor the expression.

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Problem 131

Explain how to find the GCF of $32 a^{3}$ and $16 a^{2}$

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Problem 132

Explain this diagram.
$$
\text { Multiplication } \longrightarrow
$$
$$
\begin{aligned}
3 x^{2}\left(5 x^{2}-6 x+4\right) &=15 x^{4}-18 x^{3}+12 x^{2} \\
&\longleftarrow{ Factoring }
\end{aligned}
$$

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Problem 133

Explain how factorizations of polynomials are checked. Give an example.

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Problem 134

Explain the error.
Factor out the GCF: $$30 a^{3}-12 a^{2}=6 a\left(5 a^{2}-2 a\right)$$

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Problem 135

A college student's good grades earned her a student discount on her car insurance premium. What was the percent of decrease, to the nearest percent, if her annual premium was lowered from $\$ 1,050$ to $\$ 925 ?$

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Problem 136

A student has test scores of $68 \%, 75 \%$ and $79 \%$ in a government class. What must she score on the last exam to earn a $\mathrm{B}(80 \%$ or better) in the course?

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Problem 137

$$
\text { Factor: } 6 x^{4 m} y^{n}+21 x^{3 m} y^{2 n}-15 x^{2 m} y^{3 n}
$$

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Problem 138

$$
\text { Factor: } 6 x^{4 m} y^{n}+21 x^{3 m} y^{2 n}-15 x^{2 m} y^{3 n}
$$

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