Intermediate Algebra

Lynn Marecek

Chapter 5

Polynomials and Polynomial Functions - all with Video Answers

Educators

Section 1

Add and Subtract Polynomials

Problem 1

In the following exercises, determine if the polynomial is a monomial, binomial, trinomial, or other polynomial. Then, indicate the degree of the polynomial.
(a) $47 x^{5}-17 x^{2} y^{3}+y^{2}$
(b) $5 c^{3}+11 c^{2}-c-8$
(c) $\frac{5}{9} a b+\frac{1}{3} b$
(d) 4
(e) $4 p q+17$

Nick Johnson

Nick Johnson

Numerade Educator

Problem 2

In the following exercises, determine if the polynomial is a monomial, binomial, trinomial, or other polynomial. Then, indicate the degree of the polynomial.
(a) $x^{2}-y^{2}$
(b) $-13 c^{4}$
(c) $a^{2}+2 a b-7 b^{2}$
(d) $4 x^{2} y^{2}-3 x y+8$
(e) 19

Nick Johnson

Numerade Educator

Problem 3

In the following exercises, determine if the polynomial is a monomial, binomial, trinomial, or other polynomial. Then, indicate the degree of the polynomial.
(a) $8 y-5 x$
(b) $y^{2}-5 y z-6 z^{2}$
(c) $y^{3}-8 y^{2}+2 y-16$
(d) $81 a b^{4}-24 a^{2} b^{2}+3 b$
(c) -18

Nicole Hoffman

Numerade Educator

Problem 4

In the following exercises, determine if the polynomial is a monomial, binomial, trinomial, or other polynomial. Then, indicate the degree of the polynomial.
(a) $11 y^{2}$
(b) -73
(c) $6 x^{2}-3 x y+4 x-2 y+y^{2}$
(d) $4 y^{2}+17 z^{2}$
(c) $5 c^{3}+11 c^{2}-c-8$

Nicole Hoffman

Numerade Educator

Problem 5

In the following exercises, determine if the polynomial is a monomial, binomial, trinomial, or other polynomial. Then, indicate the degree of the polynomial.
(a) $5 a^{2}+12 a b-7 b^{2}$
(b) $18 x y^{2} z$
(c) $5 x+2$
(d) $y^{3}-8 y^{2}+2 y-16$
(c) -24

Nicole Hoffman

Numerade Educator

Problem 6

In the following exercises, determine if the polynomial is a monomial, binomial, trinomial, or other polynomial. Then, indicate the degree of the polynomial.
(a) $9 y^{3}-10 y^{2}+2 y-6$
(b) $-12 p^{3} q$
(c) $a^{2}+9 a b+18 b^{2}$
(a) $20 x^{2} y^{2}-10 a^{2} b^{2}+30$
(e) 17

Nicole Hoffman

Numerade Educator

Problem 7

In the following exercises, determine if the polynomial is a monomial, binomial, trinomial, or other polynomial. Then, indicate the degree of the polynomial.
(a) $14 s-29 t$
(b) $z^{2}-5 z-6$
(c) $y^{3}-8 y^{2} z+2 y z^{2}-16 z^{3}$
(a) $23 a b^{2}-14$
(c) -3

Nicole Hoffman

Numerade Educator

Problem 8

In the following exercises, determine if the polynomial is a monomial, binomial, trinomial, or other polynomial. Then, indicate the degree of the polynomial.
(a) $15 x y$
(b) 15
(c) $6 x^{2}-3 x y+4 x-2 y+y^{2}$
(d) $10 p-9 q$
(e) $m^{4}+4 m^{3}+6 m^{2}+4 m+1$

Nicole Hoffman

Numerade Educator

Problem 9

In the following exercises, add or subtract the monomials.
(a) $7 x^{2}+5 x^{2}$
(b) $4 a-9 a$

Nick Johnson

Numerade Educator

Problem 10

In the following exercises, add or subtract the monomials.
(a) $4 y^{3}+6 y^{3}$
(b) $-y-5 y$

Nick Johnson

Numerade Educator

Problem 11

In the following exercises, add or subtract the monomials.
(a) $-12 w+18 w$
(b) $7 x^{2} y-\left(-12 x^{2} y\right)$

Nick Johnson

Numerade Educator

Problem 12

In the following exercises, add or subtract the monomials.
(a) $-3 m+9 m$
(b) $15 y z^{2}-\left(-8 y z^{2}\right)$

Nick Johnson

Numerade Educator

Problem 13

In the following exercises, add or subtract the monomials.
$7 x^{2}+5 x^{2}+4 a-9 a$

Nick Johnson

Numerade Educator

Problem 14

In the following exercises, add or subtract the monomials.
$4 y^{3}+6 y^{3}-y-5 y$

Nick Johnson

Numerade Educator

Problem 15

In the following exercises, add or subtract the monomials.
$-12 w+18 w+7 x^{2} y-\left(-12 x^{2} y\right)$

Nick Johnson

Numerade Educator

Problem 16

In the following exercises, add or subtract the monomials.
$-3 m+9 m+15 y z^{2}-\left(-8 y z^{2}\right)$

Nick Johnson

Numerade Educator

Problem 17

In the following exercises, add or subtract the monomials.
(a) $-5 b-17 b$
(b) $3 x y-(-8 x y)+5 x y$

Nick Johnson

Numerade Educator

Problem 18

In the following exercises, add or subtract the monomials.
(a) $-10 x-35 x$
(b) $17 m n^{2}-\left(-9 m n^{2}\right)+3 m n^{2}$

Nick Johnson

Numerade Educator

Problem 19

In the following exercises, add or subtract the monomials.
(a) $12 a+5 b-22 a$
(b) $p q^{2}-4 p-3 q^{2}$

Nick Johnson

Numerade Educator

Problem 20

In the following exercises, add or subtract the monomials.
(a) $14 x-3 y-13 x$
(b) $a^{2} b-4 a-5 a b^{2}$

Nick Johnson

Numerade Educator

Problem 21

In the following exercises, add or subtract the monomials.
(a) $2 a^{2}+b^{2}-6 a^{2}$
(b) $x^{2} y-3 x+7 x y^{2}$

Nick Johnson

Numerade Educator

Problem 22

In the following exercises, add or subtract the monomials.
(a) $5 u^{2}+4 v^{2}-6 u^{2}$
(b) $12 \mathrm{a}+8 b$

Nick Johnson

Numerade Educator

Problem 23

In the following exercises, add or subtract the monomials.
(a) $x y^{2}-5 x-5 y^{2}$
(b) $19 y+5 z$

Nick Johnson

Numerade Educator

Problem 24

In the following exercises, add or subtract the monomials.
$12 a+5 b-22 a+p q^{2}-4 p-3 q^{2}$

Nick Johnson

Numerade Educator

Problem 25

In the following exercises, add or subtract the monomials.
$14 x-3 y-13 x+a^{2} b-4 a-5 a b^{2}$

Nick Johnson

Numerade Educator

Problem 26

In the following exercises, add or subtract the monomials.
$2 a^{2}+b^{2}-6 a^{2}+x^{2} y-3 x+7 x y^{2}$

Nick Johnson

Numerade Educator

Problem 27

In the following exercises, add or subtract the monomials.
$5 u^{2}+4 v^{2}-6 u^{2}+12 a+8 b$

Nick Johnson

Numerade Educator

Problem 28

In the following exercises, add or subtract the monomials.
$x y^{2}-5 x-5 y^{2}+19 y+5 z$

Nick Johnson

Numerade Educator

Problem 29

In the following exercises, add or subtract the monomials.
Add: $4 a,-3 b,-8 a$

Cory Kuzinski

Numerade Educator

Problem 30

In the following exercises, add or subtract the monomials.
Add: $4 \mathrm{x}, 3 y,-3 x$

Cory Kuzinski

Numerade Educator

Problem 31

In the following exercises, add or subtract the monomials.
Subtract $5 x^{6}$ from $-12 x^{6}$

Cory Kuzinski

Numerade Educator

Problem 32

In the following exercises, add or subtract the monomials.
Subtract $2 p^{4}$ from $-7 p^{4}$

Cory Kuzinski

Numerade Educator

Problem 33

In the following exercises, add the polynomials.
$\left(5 y^{2}+12 y+4\right)+\left(6 y^{2}-8 y+7\right)$

Nick Johnson

Numerade Educator

Problem 34

In the following exercises, add the polynomials.
$\left(4 y^{2}+10 y+3\right)+\left(8 y^{2}-6 y+5\right)$

Nick Johnson

Numerade Educator

Problem 35

In the following exercises, add the polynomials.
$\left(x^{2}+6 x+8\right)+\left(-4 x^{2}+11 x-9\right)$

Nick Johnson

Numerade Educator

Problem 36

In the following exercises, add the polynomials.
$\left(y^{2}+9 y+4\right)+\left(-2 y^{2}-5 y-1\right)$

Nick Johnson

Numerade Educator

Problem 37

In the following exercises, add the polynomials.
$\left(8 x^{2}-5 x+2\right)+\left(3 x^{2}+3\right)$

Nick Johnson

Numerade Educator

Problem 38

In the following exercises, add the polynomials.
$\left(7 x^{2}-9 x+2\right)+\left(6 x^{2}-4\right)$

Nick Johnson

Numerade Educator

Problem 39

In the following exercises, add the polynomials.
$\left(5 a^{2}+8\right)+\left(a^{2}-4 a-9\right)$

Nick Johnson

Numerade Educator

Problem 40

In the following exercises, add the polynomials.
$\left(p^{2}-6 p-18\right)+\left(2 p^{2}+11\right)$

Nick Johnson

Numerade Educator

Problem 41

In the following exercises, subtract the polynomials.
$\left(4 m^{2}-6 m-3\right)-\left(2 m^{2}+m-7\right)$

Nick Johnson

Numerade Educator

Problem 42

In the following exercises, subtract the polynomials.
$\left(3 b^{2}-4 b+1\right)-\left(5 b^{2}-b-2\right)$

Nick Johnson

Numerade Educator

Problem 43

In the following exercises, subtract the polynomials.
$\left(a^{2}+8 a+5\right)-\left(a^{2}-3 a+2\right)$

Nick Johnson

Numerade Educator

Problem 44

In the following exercises, subtract the polynomials.
$\left(b^{2}-7 b+5\right)-\left(b^{2}-2 b+9\right)$

Nick Johnson

Numerade Educator

Problem 45

In the following exercises, subtract the polynomials.
$\left(12 s^{2}-15 s\right)-(s-9)$

Nick Johnson

Numerade Educator

Problem 45

In the following exercises, subtract the polynomials.
$\left(12 s^{2}-15 s\right)-(s-9)$

Nick Johnson

Numerade Educator

Problem 46

In the following exercises, subtract the polynomials.
$\left(10 r^{2}-20 r\right)-(r-8)$

Nick Johnson

Numerade Educator

Problem 47

Subtract $\left(9 x^{2}+2\right)$ from $\left(12 x^{2}-x+6\right)$

Oluwaseyitan Balogun

Oluwaseyitan Balogun

Numerade Educator

Problem 48

Subtract $\left(5 y^{2}-y+12\right)$ from $\left(10 y^{2}-8 y-20\right)$

Oluwaseyitan Balogun

Oluwaseyitan Balogun

Numerade Educator

Problem 49

Subtract $\left(7 w^{2}-4 w+2\right)$ from $\left(8 w^{2}-w+6\right)$

Oluwaseyitan Balogun

Oluwaseyitan Balogun

Numerade Educator

Problem 50

Subtract $\left(5 x^{2}-x+12\right)$ from $\left(9 x^{2}-6 x-20\right)$

Oluwaseyitan Balogun

Oluwaseyitan Balogun

Numerade Educator

Problem 51

Find the difference of $\left(w^{2}+w-42\right)$ and $\left(w^{2}-10 w+24\right)$

Oluwaseyitan Balogun

Oluwaseyitan Balogun

Numerade Educator

Problem 52

Find the difference of $\left(z^{2}-3 z-18\right)$ and $\left(z^{2}+5 z-20\right)$

Oluwaseyitan Balogun

Oluwaseyitan Balogun

Numerade Educator

Problem 53

$\left(7 x^{2}-2 x y+6 y^{2}\right)+\left(3 x^{2}-5 x y\right)$

Nick Johnson

Numerade Educator

Problem 54

$\left(-5 x^{2}-4 x y-3 y^{2}\right)+\left(2 x^{2}-7 x y\right)$

Nick Johnson

Numerade Educator

Problem 55

$\left(7 m^{2}+m n-8 n^{2}\right)+\left(3 m^{2}+2 m n\right)$

Nick Johnson

Numerade Educator

Problem 56

$\left(2 r^{2}-3 r s-2 s^{2}\right)+\left(5 r^{2}-3 r s\right)$

Nick Johnson

Numerade Educator

Problem 57

$\left(a^{2}-b^{2}\right)-\left(a^{2}+3 a b-4 b^{2}\right)$

Nick Johnson

Numerade Educator

Problem 58

$\left(m^{2}+2 n^{2}\right)-\left(m^{2}-8 m n-n^{2}\right)$

Nick Johnson

Numerade Educator

Problem 59

$\left(p^{3}-3 p^{2} q\right)+\left(2 p q^{2}+4 q^{3}\right)-\left(3 p^{2} q+p q^{2}\right)$

Nick Johnson

Numerade Educator

Problem 60

$\left(a^{3}-2 a^{2} b\right)+\left(a b^{2}+b^{3}\right)-\left(3 a^{2} b+4 a b^{2}\right)$

Nick Johnson

Numerade Educator

Problem 61

$\left(x^{3}-x^{2} y\right)-\left(4 x y^{2}-y^{3}\right)+\left(3 x^{2} y-x y^{2}\right)$

Nick Johnson

Numerade Educator

Problem 62

$\left(x^{3}-2 x^{2} y\right)-\left(x y^{2}-3 y^{3}\right)-\left(x^{2} y-4 x y^{2}\right)$

Nick Johnson

Numerade Educator

Problem 63

In the following exercises, find the function values for each polynomial function.
For the function $f(x)=8 x^{2}-3 x+2,$ find:
(a) $f(5)$
(b) $f(-2)$
(c) $f(0)$

Nick Johnson

Numerade Educator

Problem 64

In the following exercises, find the function values for each polynomial function.
For the function $f(x)=5 x^{2}-x-7$, find:
(a) $f(-4)$ (b) $f(1) \odot f(0)$

Nick Johnson

Numerade Educator

Problem 65

In the following exercises, find the function values for each polynomial function.
For the function $g(x)=4-36 x$, find:
(a) $g(3)$ (b) $g(0)$
(c) $g(-1)$

Nick Johnson

Numerade Educator

Problem 66

In the following exercises, find the function values for each polynomial function.
For the function $g(x)=16-36 x^{2}$, find:
(a) $g(-1)$ (b) $g(0)$
(c) $g(2)$

Nick Johnson

Numerade Educator

Problem 67

A painter drops a brush from a platform 75 feet high. The polynomial function $h(t)=-16 t^{2}+75$ gives the height of the brush $t$ seconds after it was dropped. Find the height after $t=2$ seconds.

Nick Johnson

Numerade Educator

Problem 68

A girl drops a ball off a 200 -foot cliff into the ocean. The polynomial $h(t)=-16 t^{2}+200$ gives the height of the ball, in feet, $t$ seconds after it is dropped. Find the height after $t=3$ seconds.

Nick Johnson

Numerade Educator

Problem 69

A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of $p$ dollars each is given by the polynomial function $R(p)=-4 p^{2}+420 p$. Find the revenue
received when $p=60$ dollars.

Nick Johnson

Numerade Educator

Problem 69

A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of $p$ dollars each is given by the polynomial function $R(p)=-4 p^{2}+420 p$. Find the revenue
received when $p=60$ dollars.

Nick Johnson

Numerade Educator

Problem 70

A manufacturer of the latest basketball shoes has found that the revenue received from selling the shoes at a cost of $p$ dollars each is given by the polynomial $R(p)=-4 p^{2}+420 p$. Find the revenue received
when $p=90$ dollars.

Nick Johnson

Numerade Educator

Problem 71

The polynomial $C(x)=6 x^{2}+90 x$ gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side $x$ feet and height 6 feet. Find the cost of producing a box with $x=4$ feet.

Nick Johnson

Numerade Educator

Problem 72

The polynomial $C(x)=6 x^{2}+90 x$ gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side $x$ feet and height 4 feet. Find the cost of producing a box with $x=6$ feet.

Nick Johnson

Numerade Educator

Problem 73

$\begin{array}{llll}\text { In each example, find } & (f+g)(x) & (b)(f+g)(2) & (c)(f-g)(x) & \text { (d) }(f-g)(-3) .\end{array}$
$f(x)=2 x^{2}-4 x+1$ and $g(x)=5 x^{2}+8 x+3$

Nick Johnson

Numerade Educator

Problem 74

$\begin{array}{llll}\text { In each example, find } & (f+g)(x) & (b)(f+g)(2) & (c)(f-g)(x) & \text { (d) }(f-g)(-3) .\end{array}$
$f(x)=4 x^{2}-7 x+3$ and $g(x)=4 x^{2}+2 x-1$

Nicole Hoffman

Numerade Educator

Problem 75

$\begin{array}{llll}\text { In each example, find } & (f+g)(x) & (b)(f+g)(2) & (c)(f-g)(x) & \text { (d) }(f-g)(-3) .\end{array}$
$f(x)=3 x^{3}-x^{2}-2 x+3$ and $g(x)=3 x^{3}-7 x$

Nicole Hoffman

Numerade Educator

Problem 76

$\begin{array}{llll}\text { In each example, find } & (f+g)(x) & (b)(f+g)(2) & (c)(f-g)(x) & \text { (d) }(f-g)(-3) .\end{array}$
$f(x)=5 x^{3}-x^{2}+3 x+4$ and $g(x)=8 x^{3}-1$

Nicole Hoffman

Numerade Educator

Problem 77

Using your own words, explain the difference between a monomial, a binomial, and a trinomial.

Cory Kuzinski

Numerade Educator

Problem 78

Using your own words, explain the difference between a polynomial with five terms and a polynomial with a degree of 5 .

Oluwaseyitan Balogun

Oluwaseyitan Balogun

Numerade Educator

Problem 79

Ariana thinks the sum $6 y^{2}+5 y^{4}$ is $11 y^{6}$. What is wrong with her reasoning?

Oluwaseyitan Balogun

Oluwaseyitan Balogun

Numerade Educator

Problem 80

Is every trinomial a second degree polynomial? If not, give an example.

Nick Johnson

Numerade Educator