Algebra for College Students

Margaret L. Lial, John Hornsby, Terry McGinnis

Chapter 11

Polynomial and Rational Functions - all with Video Answers

Educators

Section 1

Zeros of Polynomial Functions (I)

Problem 1

Choose the letter of the correct setup to perform synthetic division on the indicated quotient.
$$
\frac{x^{2}+3 x-6}{x-2}
$$
A. $- 2 )\overline{ 1 3 - 6 }$
B. $- 2 )\overline { - 1 } - 3 6$
C. $2 )\overline { 1 1 3 - 6 }$
D. $2 )\overline { - 1 - 3 6 }$

Aman Gupta

Aman Gupta

Numerade Educator

Problem 2

Choose the letter of the correct setup to perform synthetic division on the indicated quotient.
$$
\frac{x^{3}-3 x^{2}+2}{x-1}
$$
A. $1 )\overline { 1 - 3 2 }$
B. $- 1 )\overline { 1 - 3 2 }$
C. $ 1 )\overline{ 1 - 3 0 2 }$
D. $- 1 )\overline{ -1 3 0 2 }$

Aman Gupta

Numerade Educator

Problem 3

Fill in each blank with the appropriate response.
$$
\begin{aligned}3\\
\text { 5)} \overline{19}\\
\frac{15}{4}
\end{aligned}
$$
can be written
$$
19=5 \cdot \quad +
$$

Stephanie Carter

Stephanie Carter

Numerade Educator

Problem 4

Fill in each blank with the appropriate response.
$$
\begin{array}{c}
\quad \quad\quad\quad\quad x+3 \\
x - 1 )\overline{ x ^ { 2 } + 2 x + 3 } \\
x^{2}-x \\\hline \quad\quad\quad\quad{3 x}+3 \\
\quad\quad\quad\quad 3 x-3\\ \hline \quad\quad\quad\quad\quad\quad{6}
\end{array}
$$
can be written $x^{2}+2 x+3=$ $(x-1)$ (____) + _____

Ruby P

Numerade Educator

Problem 5

Fill in each blank with the appropriate response.
To perform the division
$$
x - 3 )\overline { x ^ { 3 } + 6 x ^ { 2 } + 2 x }
$$
using synthetic division, we begin by writing the following.
_____$ )\overline { 1\quad \quad 2 \quad \quad } $

Ruby P

Numerade Educator

Problem 6

Fill in each blank with the appropriate response.
Consider the following function.
$$
\begin{array}{l}
f(x)=2 x^{4}+6 x^{3}-5 x^{2}+3 x+8 \\
f(x)=(x-2)\left(2 x^{3}+10 x^{2}+15 x+33\right)+74
\end{array}
$$
By inspection, we can state that $f(2)=$ ____.

Brandon Cleary

Numerade Educator

Problem 7

A student attempted to divide
$$
4 x^{3}+2 x^{2}+6 \text { by } x+2
$$
synthetically by setting up the division as follows
$$- 2)\overline{ 4 26}$$
This is incorrect. WHAT WENT WRONG? Give the correct setup and the answer.

James Kiss

Numerade Educator

Problem 8

A student attempted to divide
$$
4 x^{3}+2 x^{2}+6 x \text { by } x+2
$$
synthetically by setting up the division as follows.
$$- 2)\overline{ 4 26}$$
This is incorrect. WHAT WENT WRONG? Give the correct setup and the answer.

James Kiss

Numerade Educator

Problem 9

Use synthetic division to divide.
$$
\frac{x^{2}-6 x+5}{x-1}
$$

Aman Gupta

Numerade Educator

Problem 10

Use synthetic division to divide.
$$
x^{2}+4 x-21
$$

Aman Gupta

Numerade Educator

Problem 11

Use synthetic division to divide.
$$
\frac{4 x^{2}+19 x-5}{x+5}
$$

Aman Gupta

Numerade Educator

Problem 12

Use synthetic division to divide.
$$
\frac{3 x^{2}+5 x-12}{x+3}
$$

Aman Gupta

Numerade Educator

Problem 13

Use synthetic division to divide.
$$
\frac{2 x^{2}+8 x+13}{x+2}
$$

Aman Gupta

Numerade Educator

Problem 14

Use synthetic division to divide.
$$
\frac{4 x^{2}-5 x-20}{x-4}
$$

Aman Gupta

Numerade Educator

Problem 15

Use synthetic division to divide.
$$
\frac{4 x^{3}-3 x^{2}+2 x-3}{x-1}
$$

Aman Gupta

Numerade Educator

Problem 16

Use synthetic division to divide.
$$
\frac{5 x^{3}-6 x^{2}+3 x+14}{x+1}
$$

Aman Gupta

Numerade Educator

Problem 17

Use synthetic division to divide.
$$
\frac{x^{3}+2 x^{2}-4 x+3}{x-4}
$$

Aman Gupta

Numerade Educator

Problem 18

Use synthetic division to divide.
$$
\frac{x^{3}-3 x^{2}+5 x-1}{x-5}
$$

Aman Gupta

Numerade Educator

Problem 19

Use synthetic division to divide.
$$
\frac{2 x^{5}-2 x^{3}+3 x^{2}-24 x-2}{x-2}
$$

Aman Gupta

Numerade Educator

Problem 20

Use synthetic division to divide.
$$
\frac{3 x^{5}+x^{4}-84 x^{2}-12 x+3}{x-3}
$$

Aman Gupta

Numerade Educator

Problem 21

Use synthetic division to divide.
$$
\frac{-3 x^{5}-3 x^{4}+5 x^{3}-6 x^{2}+3}{x+1}
$$

Aman Gupta

Numerade Educator

Problem 22

Use synthetic division to divide.
$$
\frac{-3 x^{5}+2 x^{4}-5 x^{3}-6 x^{2}-1}{x+2}
$$

Aman Gupta

Numerade Educator

Problem 23

Use synthetic division to divide.
$$
\frac{x^{5}+x^{4}+x^{3}+x^{2}+x+3}{x+1}
$$

Aman Gupta

Numerade Educator

Problem 24

Use synthetic division to divide.
$$
\frac{x^{5}-x^{4}+x^{3}-x^{2}+x-2}{x-1}
$$

Aman Gupta

Numerade Educator

Problem 25

Express each polynomial function in the form $f(x)=(x-k) q(x)+r$ for the given value of k.
$$
f(x)=2 x^{3}+x^{2}+x-8 ; \quad k=-1
$$

Aman Gupta

Numerade Educator

Problem 26

Express each polynomial function in the form $f(x)=(x-k) q(x)+r$ for the given value of k.
$$
f(x)=2 x^{3}+3 x^{2}-6 x+1 ; \quad k=-4
$$

Aman Gupta

Numerade Educator

Problem 27

Express each polynomial function in the form $f(x)=(x-k) q(x)+r$ for the given value of k.
$$
f(x)=-x^{3}+2 x^{2}+4 ; \quad k=-2
$$

Aman Gupta

Numerade Educator

Problem 28

Express each polynomial function in the form $f(x)=(x-k) q(x)+r$ for the given value of k.
$$
f(x)=-2 x^{3}+6 x^{2}+5 ; \quad k=2
$$

Aman Gupta

Numerade Educator

Problem 29

Express each polynomial function in the form $f(x)=(x-k) q(x)+r$ for the given value of k.
$$
f(x)=4 x^{4}-3 x^{3}-20 x^{2}-x ; \quad k=3
$$

Aman Gupta

Numerade Educator

Problem 30

Express each polynomial function in the form $f(x)=(x-k) q(x)+r$ for the given value of k.
$$
f(x)=2 x^{4}+x^{3}-15 x^{2}+3 x ; \quad k=-3
$$

Aman Gupta

Numerade Educator

Problem 31

For each polynomial function, use the remainder theorem and synthetic division to find $f(k) .$
$$
f(x)=x^{2}-4 x+5 ; \quad k=3
$$

Aman Gupta

Numerade Educator

Problem 32

For each polynomial function, use the remainder theorem and synthetic division to find $f(k) .$
$$
f(x)=x^{2}+5 x+6 ; \quad k=-2
$$

Aman Gupta

Numerade Educator

Problem 33

For each polynomial function, use the remainder theorem and synthetic division to find $f(k) .$
$$
f(x)=2 x^{2}-3 x-3 ; \quad k=2
$$

Aman Gupta

Numerade Educator

Problem 34

For each polynomial function, use the remainder theorem and synthetic division to find $f(k) .$
$$
f(x)=-x^{3}+8 x^{2}+63 ; \quad k=4
$$

Aman Gupta

Numerade Educator

Problem 35

For each polynomial function, use the remainder theorem and synthetic division to find $f(k) .$
$$
f(x)=x^{3}-4 x^{2}+2 x+1 ; \quad k=-1
$$

Aman Gupta

Numerade Educator

Problem 36

For each polynomial function, use the remainder theorem and synthetic division to find $f(k) .$
$$
f(x)=2 x^{3}-3 x^{2}-5 x+4 ; \quad k=2
$$

Aman Gupta

Numerade Educator

Problem 37

For each polynomial function, use the remainder theorem and synthetic division to find $f(k) .$
$$
f(x)=2 x^{5}-10 x^{3}-19 x^{2} ; \quad k=3
$$

Aman Gupta

Numerade Educator

Problem 38

For each polynomial function, use the remainder theorem and synthetic division to find $f(k) .$
$$
f(x)=x^{4}-6 x^{3}+9 x^{2}-3 x ; \quad k=4
$$

Aman Gupta

Numerade Educator

Problem 39

For each polynomial function, use the remainder theorem and synthetic division to find $f(k) .$
$$
f(x)=x^{2}-5 x+1 ; \quad k=2+i
$$

Aman Gupta

Numerade Educator

Problem 40

For each polynomial function, use the remainder theorem and synthetic division to find $f(k) .$
$$
f(x)=x^{2}-x+3 ; \quad k=3-2 i
$$

Aman Gupta

Numerade Educator

Problem 41

For each polynomial function, use the remainder theorem and synthetic division to find $f(k) .$
$$
f(x)=9 x^{3}-6 x^{2}+x ; \quad k=\frac{1}{3}
$$

Aman Gupta

Numerade Educator

Problem 42

For each polynomial function, use the remainder theorem and synthetic division to find $f(k) .$
$$
f(x)=6 x^{3}-31 x^{2}-15 x ; \quad k=-\frac{1}{2}
$$

Aman Gupta

Numerade Educator

Problem 43

Use synthetic division to determine whether the given number is a zero of the polynomial function.
$$
3 ; \quad f(x)=2 x^{3}-6 x^{2}-9 x+27
$$

Aman Gupta

Numerade Educator

Problem 44

Use synthetic division to determine whether the given number is a zero of the polynomial function.
$$
-6 ; \quad f(x)=2 x^{3}+9 x^{2}-16 x+12
$$

Aman Gupta

Numerade Educator

Problem 45

Use synthetic division to determine whether the given number is a zero of the polynomial function.
$$
-5 ; \quad f(x)=x^{3}+7 x^{2}+10 x
$$

Aman Gupta

Numerade Educator

Problem 46

Use synthetic division to determine whether the given number is a zero of the polynomial function.
$$
-2 ; \quad f(x)=x^{3}-7 x^{2}-18 x
$$

Aman Gupta

Numerade Educator

Problem 47

Use synthetic division to determine whether the given number is a zero of the polynomial function.
$$
\frac{2}{5} ; \quad f(x)=5 x^{4}+2 x^{3}-x+15
$$

Aman Gupta

Numerade Educator

Problem 48

Use synthetic division to determine whether the given number is a zero of the polynomial function.
$$
\frac{1}{2} ; \quad f(x)=2 x^{4}-3 x^{2}+4
$$

Aman Gupta

Numerade Educator

Problem 49

Use synthetic division to determine whether the given number is a zero of the polynomial function.
$$
2-i ; \quad f(x)=x^{2}+3 x+4
$$

Aman Gupta

Numerade Educator

Problem 50

Use synthetic division to determine whether the given number is a zero of the polynomial function.
$$
1-2 i ; \quad f(x)=x^{2}-3 x+5
$$

Aman Gupta

Numerade Educator

Problem 51

We have seen the close connection between polynomial division and writing a quotient of polynomials in lowest terms after factoring the numerator. We can also show a connection between dividing one polynomial by another and factoring the first polynomial. letting
$$
f(x)=2 x^{2}+5 x-12
$$
Factor $f(x)$

James Kiss

Numerade Educator

Problem 52

We have seen the close connection between polynomial division and writing a quotient of polynomials in lowest terms after factoring the numerator. We can also show a connection between dividing one polynomial by another and factoring the first polynomial. letting
$$
f(x)=2 x^{2}+5 x-12
$$
Solve $f(x)=0$

Demi Nelson

Numerade Educator

Problem 53

We have seen the close connection between polynomial division and writing a quotient of polynomials in lowest terms after factoring the numerator. We can also show a connection between dividing one polynomial by another and factoring the first polynomial. letting
$$
f(x)=2 x^{2}+5 x-12
$$
Evaluate $f(-4)$

Demi Nelson

Numerade Educator

Problem 54

We have seen the close connection between polynomial division and writing a quotient of polynomials in lowest terms after factoring the numerator. We can also show a connection between dividing one polynomial by another and factoring the first polynomial. letting
$$
f(x)=2 x^{2}+5 x-12
$$
Evaluate $f\left(\frac{3}{2}\right)$.

Demi Nelson

Numerade Educator

Problem 55

We have seen the close connection between polynomial division and writing a quotient of polynomials in lowest terms after factoring the numerator. We can also show a connection between dividing one polynomial by another and factoring the first polynomial. letting
$$
f(x)=2 x^{2}+5 x-12.
$$
Complete the following sentence: If $f(a)=0,$ then $x-$ ______is a factor of $f(x)$.

Demi Nelson

Numerade Educator

Problem 56

We have seen the close connection between polynomial division and writing a quotient of polynomials in lowest terms after factoring the numerator. We can also show a connection between dividing one polynomial by another and factoring the first polynomial. letting
$$
f(x)=2 x^{2}+5 x-12.
$$
Use the conclusion reached in to determine whether $x-3$ is a factor of $g(x)=3 x^{3}-4 x^{2}-17 x+6 .$ If so, factor $g(x)$ completely.

Gregory Higby

Numerade Educator