Introductory Algebra for College Students

Problem 3

Find the greatest common factor of each list of monomials.
$12 x^{2}$ and $8 x$

Erika Bustos

Numerade Educator

00:25

Problem 4

Find the greatest common factor of each list of monomials.
$$20 x^{2} \text { and } 15 x$$

Erika Bustos

Numerade Educator

Problem 5

Find the greatest common factor of each list of monomials.
$$-2 x^{4} \text { and } 6 x^{3}$$

Erika Bustos

Numerade Educator

Problem 6

Find the greatest common factor of each list of monomials.
$$-3 x^{4} \text { and } 6 x^{3}$$

Erika Bustos

Numerade Educator

Problem 7

Find the greatest common factor of each list of monomials.
$$9 y^{5}, 18 y^{2}, \text { and }-3 y$$

Teresa Liang

Numerade Educator

00:25

Problem 8

Find the greatest common factor of each list of monomials.
$$10 y^{5}, 20 y^{2}, \text { and }-5 y$$

Teresa Liang

Numerade Educator

Problem 9

Find the greatest common factor of each list of monomials.
$$x y, x y^{2}, \text { and } x y^{3}$$

Erika Bustos

Numerade Educator

00:35

Problem 10

Find the greatest common factor of each list of monomials.
$$x^{2} y, 3 x^{3} y, \text { and } 6 x^{2}$$

Erika Bustos

Numerade Educator

01:32

Problem 11

Find the greatest common factor of each list of monomials.
$$16 x^{5} y^{4}, 8 x^{6} y^{3}, \text { and } 20 x^{4} y^{5}$$

Erika Bustos

Numerade Educator

01:19

Problem 12

Find the greatest common factor of each list of monomials.
$$18 x^{5} y^{4}, 6 x^{6} y^{3}, \text { and } 12 x^{4} y^{5}$$

Erika Bustos

Numerade Educator

00:22

Problem 13

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$8 x+8$$

Teresa Liang

Numerade Educator

00:19

Problem 14

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$9 x+9$$

Teresa Liang

Numerade Educator

00:18

Problem 15

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$4 y-4$$

Teresa Liang

Numerade Educator

00:18

Problem 16

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$5 y-5$$

Teresa Liang

Numerade Educator

00:16

Problem 17

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$5 x+30$$

Teresa Liang

Numerade Educator

00:21

Problem 18

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$10 x+30$$

Teresa Liang

Numerade Educator

00:21

Problem 19

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$30 x-12$$

Teresa Liang

Numerade Educator

Problem 20

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$32 x-24$$

Teresa Liang

Numerade Educator

00:17

Problem 21

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$x^{2}+5 x$$

Teresa Liang

Numerade Educator

00:16

Problem 22

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$x^{2}+6 x$$

Teresa Liang

Numerade Educator

Problem 23

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$18 y^{2}+12$$

Teresa Liang

Numerade Educator

00:28

Problem 24

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$20 y^{2}+15$$

Teresa Liang

Numerade Educator

00:35

Problem 25

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$14 x^{3}+21 x^{2}$$

Teresa Liang

Numerade Educator

00:46

Problem 26

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$6 x^{3}+15 x^{2}$$

Teresa Liang

Numerade Educator

Problem 27

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$13 y^{2}-25 y$$

Teresa Liang

Numerade Educator

00:24

Problem 28

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$11 y^{2}-30 y$$

Teresa Liang

Numerade Educator

00:28

Problem 29

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$9 y^{4}+27 y^{6}$$

Teresa Liang

Numerade Educator

Problem 30

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$10 y^{4}+15 y^{6}$$

Teresa Liang

Numerade Educator

Problem 31

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$8 x^{2}-4 x^{4}$$

Teresa Liang

Numerade Educator

Problem 32

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$12 x^{2}-4 x^{4}$$

Teresa Liang

Numerade Educator

Problem 33

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$12 y^{2}+16 y-8$$

Teresa Liang

Numerade Educator

Problem 34

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$15 y^{2}-3 y+9$$

Teresa Liang

Numerade Educator

00:42

Problem 35

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$9 x^{4}+18 x^{3}+6 x^{2}$$

Teresa Liang

Numerade Educator

00:36

Problem 36

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$32 x^{4}+2 x^{3}+8 x^{2}$$

Teresa Liang

Numerade Educator

00:44

Problem 37

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$100 y^{5}-50 y^{3}+100 y^{2}$$

Teresa Liang

Numerade Educator

00:40

Problem 38

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$26 y^{5}-13 y^{3}+39 y^{2}$$

Teresa Liang

Numerade Educator

Problem 39

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$10 x-20 x^{2}+5 x^{3}$$

Teresa Liang

Numerade Educator

00:28

Problem 40

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$6 x-4 x^{2}+2 x^{3}$$

Teresa Liang

Numerade Educator

00:16

Problem 41

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$11 x^{2}-23$$

Teresa Liang

Numerade Educator

00:14

Problem 42

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$12 x^{2}-25$$

Teresa Liang

Numerade Educator

00:31

Problem 43

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$6 x^{3} y^{2}+9 x y$$

Teresa Liang

Numerade Educator

Problem 44

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$4 x^{2} y^{3}+6 x y$$

Teresa Liang

Numerade Educator

Problem 45

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$30 x^{2} y^{3}-10 x y^{2}+20 x y$$

Teresa Liang

Numerade Educator

01:08

Problem 46

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$27 x^{2} y^{3}-18 x y^{2}+45 x^{2} y$$

Teresa Liang

Numerade Educator

00:55

Problem 46

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$32 x^{3} y^{2}-24 x^{3} y-16 x^{2} y$$

Teresa Liang

Numerade Educator

00:51

Problem 48

Factor each polynomial using the greatest common factor. If there is no common factor other than 1 and the polynomial cannot be factored, so state.
$$18 x^{3} y^{2}-12 x^{3} y-24 x^{2} y$$

Teresa Liang

Numerade Educator

Problem 49

Factor each polynomial using the negative of the greatest common factor.
$$-12 x^{2}+18$$

Teresa Liang

Numerade Educator

00:32

Problem 50

Factor each polynomial using the negative of the greatest common factor.
$$-15 x^{2}+20$$

Teresa Liang

Numerade Educator

00:50

Problem 51

Factor each polynomial using the negative of the greatest common factor.
$$-8 x^{4}+32 x^{3}+16 x^{2}$$

Teresa Liang

Numerade Educator

Problem 52

Factor each polynomial using the negative of the greatest common factor.
$$-18 x^{4}+9 x^{3}+6 x^{2}$$

Teresa Liang

Numerade Educator

Problem 53

Factor each polynomial using the negative of the greatest common factor.
$$-4 a^{3} b^{2}+6 a b$$

Teresa Liang

Numerade Educator

Problem 54

Factor each polynomial using the negative of the greatest common factor.
$$-9 a^{2} b^{3}+12 a b$$

Teresa Liang

Numerade Educator

00:56

Problem 55

Factor each polynomial using the negative of the greatest common factor.
$$-12 x^{3} y^{2}-18 x^{3} y+24 x^{2} y$$

Teresa Liang

Numerade Educator

00:56

Problem 56

Factor each polynomial using the negative of the greatest common factor.
$$-24 x^{3} y^{2}-32 x^{3} y+16 x^{2} y$$

Teresa Liang

Numerade Educator

Problem 57

Factor each polynomial using the greatest common binomial factor.
$$x(x+5)+3(x+5)$$

Teresa Liang

Numerade Educator

Problem 58

Factor each polynomial using the greatest common binomial factor.
$$x(x+7)+10(x+7)$$

Teresa Liang

Numerade Educator

Problem 59

Factor each polynomial using the greatest common binomial factor.
$$x(x+2)-4(x+2)$$

Teresa Liang

Numerade Educator

Problem 60

Factor each polynomial using the greatest common binomial factor.
$$x(x+3)-8(x+3)$$

Teresa Liang

Numerade Educator

00:25

Problem 61

Factor each polynomial using the greatest common binomial factor.
$$x(y+6)-7(y+6)$$

Teresa Liang

Numerade Educator

Problem 62

Factor each polynomial using the greatest common binomial factor.
$$x(y+9)-11(y+9)$$

Teresa Liang

Numerade Educator

Problem 63

Factor each polynomial using the greatest common binomial factor.
$$3 x(x+y)-(x+y)$$

Teresa Liang

Numerade Educator

00:39

Problem 64

Factor each polynomial using the greatest common binomial factor.
$$7 x(x+y)-(x+y)$$

Teresa Liang

Numerade Educator

00:46

Problem 65

Factor each polynomial using the greatest common binomial factor.
$$4 x(3 x+1)+3 x+1$$

Teresa Liang

Numerade Educator

00:43

Problem 66

Factor each polynomial using the greatest common binomial factor.
$$5 x(2 x+1)+2 x+1$$

Teresa Liang

Numerade Educator

Problem 67

Factor each polynomial using the greatest common binomial factor.
$$7 x^{2}(5 x+4)+5 x+4$$

Teresa Liang

Numerade Educator

00:43

Problem 68

Factor each polynomial using the greatest common binomial factor.
$$9 x^{2}(7 x+2)+7 x+2$$

Teresa Liang

Numerade Educator

01:03

Problem 69

Factor by grouping.
$$x^{2}+2 x+4 x+8$$

Teresa Liang

Numerade Educator

00:50

Problem 70

Factor by grouping.
$$x^{2}+3 x+5 x+15$$

Teresa Liang

Numerade Educator

00:57

Problem 71

Factor by grouping.
$$x^{2}+3 x-5 x-15$$

Teresa Liang

Numerade Educator

02:03

Problem 72

Factor by grouping.
$$x^{2}+7 x-4 x-28$$

Pammi Eswari

Numerade Educator

00:54

Problem 73

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

Teresa Liang

Numerade Educator

00:59

Problem 74

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

Teresa Liang

Numerade Educator

00:55

Problem 75

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

Teresa Liang

Numerade Educator

00:47

Problem 76

Factor by grouping.
$$x^{3}+6 x^{2}-2 x-12$$

Teresa Liang

Numerade Educator

Problem 77

Factor by grouping.
$$x y+5 x+9 y+45$$

Teresa Liang

Numerade Educator

00:44

Problem 78

Factor by grouping.
$$x y+6 x+2 y+12$$

Teresa Liang

Numerade Educator

00:52

Problem 79

Factor by grouping.
$$x y-x+5 y-5$$

Teresa Liang

Numerade Educator

Problem 80

Factor by grouping.
$$x y-x+7 y-7$$

Teresa Liang

Numerade Educator

01:04

Problem 81

Factor by grouping.
$$3 x^{2}-6 x y+5 x y-10 y^{2}$$

Teresa Liang

Numerade Educator

01:06

Problem 82

Factor by grouping.
$$10 x^{2}-12 x y+35 x y-42 y^{2}$$

Teresa Liang

Numerade Educator

01:10

Problem 83

Factor by grouping.
$$3 x^{3}-2 x^{2}-6 x+4$$

Teresa Liang

Numerade Educator

01:00

Problem 84

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

Teresa Liang

Numerade Educator

Problem 85

Factor by grouping.
$$x^{2}-a x-b x+a b$$

Teresa Liang

Numerade Educator

00:51

Problem 86

Factor by grouping.
$$x^{2}+a x+b x+a b$$

Teresa Liang

Numerade Educator

01:10

Problem 87

Factor each polynomial.
$$24 x^{3} y^{3} z^{3}+30 x^{2} y^{2} z+18 x^{2} y z^{2}$$

Teresa Liang

Numerade Educator

01:04

Problem 88

Factor each polynomial.
$$16 x^{2} y^{2} z^{2}+32 x^{2} y z^{2}+24 x^{2} y z$$

Teresa Liang

Numerade Educator

01:09

Problem 89

Factor each polynomial.
$$x^{3}-4+3 x^{3} y-12 y$$

Teresa Liang

Numerade Educator

01:03

Problem 90

Factor each polynomial.
$$x^{3}-5+2 x^{3} y-10 y$$

Teresa Liang

Numerade Educator

01:02

Problem 91

Factor each polynomial.
$$4 x^{5}(x+1)-6 x^{3}(x+1)-8 x^{2}(x+1)$$

Teresa Liang

Numerade Educator

00:58

Problem 92

Factor each polynomial.
$$8 x^{5}(x+2)-10 x^{3}(x+2)-2 x^{2}(x+2)$$

Teresa Liang

Numerade Educator

01:10

Problem 93

Factor each polynomial.
$$3 x^{5}-3 x^{4}+x^{3}-x^{2}+5 x-5$$

Teresa Liang

Numerade Educator

01:15

Problem 94

Factor each polynomial.
$$7 x^{5}-7 x^{4}+x^{3}-x^{2}+3 x-3$$

Teresa Liang

Numerade Educator

01:32

Problem 95

Show one or more circles drawn inside a square. Write a polynomial that represents the shaded blue area in each figure. Then factor the polynomial.
(IMAGE CANNOT COPY)

Maurizia De Palma

Numerade Educator

01:32

Problem 96

Show one or more circles drawn inside a square. Write a polynomial that represents the shaded blue area in each figure. Then factor the polynomial.
(IMAGE CANNOT COPY)

Maurizia De Palma

Numerade Educator

01:20

An explosion causes debris to rise vertically with an initial velocity of 64 feet per second. The polynomial $64 x-16 x^{2}$ describes the height of the debris above the ground, in feet, after $x$ seconds.
a. Find the height of the debris after 3 seconds.
b. Factor the polynomial.
c. Use the factored form of the polynomial in part (b) to find the height of the debris after 3 seconds. Do you get the same answer as you did in part (a)? If so, does this prove that your factorization is correct? Explain.

Teresa Liang

Numerade Educator

03:38

Problem 98

An explosion causes debris to rise vertically with an initial velocity of 72 feet per second. The polynomial $72 x-16 x^{2}$ describes the height of the debris above the ground, in feet, after $x$ seconds.
a. Find the height of the debris after 4 seconds.
b. Factor the polynomial.
c. Use the factored form of the polynomial in part (b) to find the height of the debris after 4 seconds. Do you get the same answer as you did in part (a)? If so, does this prove that your factorization is correct? Explain

Kristin Mcleod

Numerade Educator

00:34

Problem 99

Write a polynomial for the length of each rectangle.
(IMAGE CANNOT COPY)

Erika Bustos

Numerade Educator

00:43

Problem 100

Write a polynomial for the length of each rectangle.
(IMAGE CANNOT COPY)

Erika Bustos

Numerade Educator

Problem 100

Write a polynomial for the length of each rectangle.
(IMAGE CANNOT COPY)

Erika Bustos

Numerade Educator

Problem 101

What is factoring?

James Kiss

Numerade Educator

Problem 102

What is a prime polynomial?

Teresa Liang

Numerade Educator

01:45

Problem 103

Explain how to find the greatest common factor of a list of terms. Give an example with your explanation.

Teresa Liang

Numerade Educator

01:15

Problem 104

Use an example and explain how to factor out the greatest common factor of a polynomial.

Teresa Liang

Numerade Educator

00:44

Problem 105

Suppose that a polynomial contains four terms and can be factored by grouping. Explain how to obtain the factorization.

Teresa Liang

Numerade Educator

00:39

Problem 106

Write a sentence that uses the word "factor" as a noun. Then write a sentence that uses the word "factor" as a verb.

Teresa Liang

Numerade Educator

01:09

Problem 107

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning.
After factoring $20 x^{3}+8 x^{2}$ and $20 x^{3}+10 x,$ I noticed that I factored the monomial $20 x^{3}$ in two different ways.

Teresa Liang

Numerade Educator

01:05

Problem 108

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning.
After I've factored a polynomial, my answer cannot always be checked by multiplication.

Teresa Liang

Numerade Educator

Problem 109

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning.
The word greatest in greatest common factor is helpful because it tells me to look for the greatest power of a variable appearing in all terms.

James Kiss

Numerade Educator

00:31

Problem 110

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning.
You grouped the polynomial's terms using different groupings than I did, yet we both obtained the same factorization.

James Kiss

Numerade Educator

Problem 111

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
since the GCF of $9 x^{3}+6 x^{2}+3 x$ is $3 x,$ it is not necessary to write the 1 when $3 x$ is factored from the last term.

Erika Bustos

Numerade Educator

01:08

Problem 112

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
$$\begin{aligned}
a(x-7)+b(7-x) &=a(x-7)+b(-1)(x-7) \\
&=a(x-7)-b(x-7) \\
&=(x-7)(a-b)
\end{aligned}$$

Erika Bustos

Numerade Educator

Problem 113

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
$$\begin{aligned}
a^{2}+b^{2} &=a^{2}+a b-a b+b^{2} \\
&=a(a+b)-b(a+b) \\
&=(a+b)(a-b)
\end{aligned}$$

James Kiss

Numerade Educator

Problem 114

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
$-4 x^{2}+12 x$ can be factored as $-4 x(x-3)$ or $4 x(-x+3)$

James Kiss

Numerade Educator

00:39

Problem 115

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
Suppose you receive $x$ dollars in January. Each month thereafter, you receive $\$ 100$ more than you received the month before. Write a factored polynomial that describes the total dollar amount you receive from January through April.

Sherrie Fenner

Numerade Educator

00:45

Problem 116

Write a polynomial that fits the given description. Do not use a polynomial that appears in this section or in the Exercise Set.
The polynomial has four terms and can be factored using a greatest common factor that has both a coefficient and a variable.

Teresa Liang

Numerade Educator

01:00

Problem 117

Write a polynomial that fits the given description. Do not use a polynomial that appears in this section or in the Exercise Set.
The polynomial has four terms and can be factored by grouping.

Teresa Liang

Numerade Educator

00:54

Problem 118

Use a graphing utility to graph each side of the equation in the same viewing rectangle. Do the graphs coincide? If so, this means that the polynomial on the left side has been factored correctly. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization.
$$-3 x-6=-3(x-2)$$

James Kiss

Numerade Educator

01:20

Problem 119

James Kiss

Numerade Educator

01:13

Problem 120

James Kiss

Numerade Educator