Deck 13: Polynomial and Synthetic Division

Question 1

Use synthetic division to divide. $\left( 4 x ^ { 3 } + 11 x ^ { 2 } - 43 x + 10 \right) \div ( x + 5 )$

A) $4 x ^ { 2 } - 11 x - 5$
B) $4 x ^ { 2 } - 9 x + 2$
C) $4 x ^ { 2 } - 3 x - 10$
D) $4 x ^ { 2 } + 19 x - 4$
E) $4 x ^ { 2 } + 6 x - 8$

Answer 1

$4 x ^ { 2 } - 9 x + 2$

Question 2

Use long division to divide. $\left( x ^ { 3 } + 64 \right) \div ( x + 4 )$

A) $x ^ { 2 } - 4 x + 16$
B) $x ^ { 2 } + 4 x - 16$
C) $x ^ { 2 } + 16$
D) $x ^ { 2 } - 16$
E) $x ^ { 2 } - 16 + \frac { 4 } { x + 4 }$

Answer 2

$x ^ { 2 } - 4 x + 16$

Question 3

Use long division to divide. $\left( x ^ { 4 } - 4 x ^ { 2 } - 2 \right) \div \left( x ^ { 2 } + x - 4 \right)$

A) $x ^ { 2 } - x + 1 - \frac { 4 } { x ^ { 2 } - x + 1 }$
B) $x ^ { 2 } - x + 1 + \frac { - 5 x + 2 } { x ^ { 2 } + x - 4 }$
C) $x ^ { 2 } - x + 1$
D) $x ^ { 2 } + x - 1$
E) $x ^ { 2 } + x - 1 + \frac { - 5 x - 4 } { x ^ { 2 } + x - 4 }$

Answer 3

$x ^ { 2 } - x + 1 + \frac { - 5 x + 2 } { x ^ { 2 } + x - 4 }$

Question 4

The amounts A (in billions of dollars)donated to support higher education in the United States from 2000 through 2007 are shown in the table,where t represents the year,with t = 0 corresponding to 2000. $$\begin{array} { | c | c | } \hline \text { Year, } t & \text { Amount, } A \\\hline 0 & 23.3 \\\hline 1 & 24.3 \\\hline 2 & 24 \\\hline 3 & 24 \\\hline 4 & 24.5 \\\hline 5 & 25.7 \\\hline 6 & 28.1 \\\hline 7 & 29.9 \\\hline\end{array}$$
Use a graphing utility to select a correct a scatter plot of the above data.

A)​
B)​
C)​
D)
E)​

Question 5

Use synthetic division to divide. $\left( x ^ { 3 } - 27 x + 54 \right) \div ( x - 3 )$

A)​ $x ^ { 2 } + 3 x - 18$
B)​ $x ^ { 2 } - 3 x - 27$
C)​ $x ^ { 2 } + 9 x + 18$
D)​ $x ^ { 2 } + 9 x - 6$
E)​ $x ^ { 2 } + 6 x + 9$

Question 6

If $f ( x ) = - 4 x ^ { 2 } + 6 x - 7$ ,use synthetic division to evaluate f (2).

A) $f ( 2 ) = - 11$
B) $f ( 2 ) = 21$
C) $f ( 2 ) = - 35$
D) $f ( 2 ) = - 3$
E) $f ( 2 ) = - 1$

Question 7

Use the Remainder Theorem and synthetic division to find the function value.Verify your answers using another method.​ $h ( x ) = x ^ { 3 } - 2 x ^ { 2 } - 5 x + 6 , \quad h ( - 8 )$ ​

A)-594
B)-592
C)-596
D)-593
E)-590

Question 8

Write $f ( x ) = x ^ { 3 } + 7 x ^ { 2 } + 13 x + 22$ in the form $f ( x ) = ( x - k ) q ( x ) + r$ when k = -5.

A) $f ( x ) = ( x + 5 ) \left( x ^ { 2 } + 2 x + 3 \right) + 7$
B) $f ( x ) = ( x + 5 ) \left( x ^ { 2 } - 2 x + 3 \right) - 7$
C) $f ( x ) = ( x - 5 ) \left( x ^ { 2 } + 2 x + 3 \right) - 7$
D) $f ( x ) = ( x - 5 ) \left( x ^ { 2 } - 2 x + 3 \right) + 7$
E) $f ( x ) = ( x + 5 ) \left( x ^ { 2 } + 2 x + 3 \right)$

Question 9

If x = -4 is a root of $x ^ { 3 } + 5 x ^ { 2 } - 16 x - 80 = 0$ ,use synthetic division to factor the polynomial completely and list all real solutions of the equation.

A) $( x + 5 ) ( x - 5 ) ( x - 4 ) ; x = - 5,5,4$
B) $( x + 5 ) ^ { 2 } ( x + 4 ) ; x = - 5 , - 4$
C) $( x + 5 ) ( x + 4 ) ^ { 2 } ; x = - 5 , - 4$
D) $( x - 5 ) ( x + 4 ) ( x - 4 ) ; x = 5 , - 4,4$
E) $( x + 5 ) ( x + 4 ) ( x - 4 ) ; x = - 5 , - 4,4$

Question 10

Use long division to divide. ​​ $$\left( x ^ { 4 } + 7 x ^ { 3 } + 10 x ^ { 2 } - x - 2 \right) \div ( x + 2 )$$ ​

A) $x ^ { 3 } + 5 x ^ { 2 } + 1$
B) $x ^ { 3 } - 5 x ^ { 2 } - 1$
C) $x ^ { 3 } + 5 x ^ { 2 } - 1$
D) $x ^ { 2 } + 5 x - 1$
E) $x ^ { 3 } - 5 x ^ { 2 } + 1$

Question 11

Use long division to divide. $$\left( x ^ { 3 } + 4 x ^ { 2 } + x + 4 \right) \div ( x + 4 )$$

A) $x ^ { 2 } + 1$
B) $$x ^ { 2 } + 8 x + 33 - \frac { 129 } { x + 4 }$$
C) $x ^ { 2 } - 4$
D) $x ^ { 2 } + 8 x + 33 + \frac { 120 } { x + 4 }$
E) $x ^ { 2 } + 8 x + 33$

Question 12

Use synthetic division to divide. $\frac { - 4 x ^ { 3 } + 23 x ^ { 2 } - 31 x + 12 } { 4 x - 3 }$

A) $4 x ^ { 2 } + 12 x + 16$
B) $x ^ { 2 } + 4 x - 4$
C) $- 16 x ^ { 2 } + 12 x + 16$
D) $- 4 x ^ { 2 } - 12 x + 16$
E) $- x ^ { 2 } + 5 x - 4$

Question 13

Select the correct graph of the following function.​ $f ( x ) = 6 x ^ { 3 } + x ^ { 2 } - 9 x + 5$

A)​
B)​
C)​
D)​
E)​

Question 14

Use long division to divide. ​​ $( 4 x + 2 ) \div ( 2 x + 1 )$ ​

A)4
B)2
C)-2
D)1
E)3

Question 15

Select the correct graph of the following function.​ $$f ( x ) = 5 x ^ { 3 } + 4 x ^ { 2 } - 8 x + 18$$

A)​
B)​
C)​
D)​
E)​

Question 16

The amounts A (in billions of dollars)donated to support higher education in the United States from 2000 through 2007 are shown in the table,where t represents the year,with t = 0 corresponding to 2000. $$\begin{array} { | c | c | } \hline \text { Year, } t & \text { Amount, } A \\\hline 0 & 30.6 \\\hline 1 & 32.3 \\\hline 2 & 34.3 \\\hline 3 & 57.7 \\\hline 4 & 38.1 \\\hline 5 & 42.8 \\\hline 6 & 48 \\\hline 7 & 52.8 \\\hline\end{array}$$
Use a graphing utility to create a scatter plot of the data.

A)​
B)
C)​
D)​
E)​

Question 17

Use the Remainder Theorem and synthetic division to find the function value.​ $$g ( x ) = 3 x ^ { 6 } + 4 x ^ { 4 } - 6 x ^ { 2 } + 4 , \quad g ( 0 )$$ ​

A)4.00
B)1.00
C)-1.00
D)6.00
E)5

Question 18

Use synthetic division to divide. $\left( 12 + 5 x ^ { 3 } - 11 x - 18 x ^ { 2 } \right) \div ( x - 4 )$

A)​ $5 x ^ { 2 } + 2 x - 3$
B)​ $5 x ^ { 2 } - 7 x + 12$
C) $5 x ^ { 2 } + 2 x + 5$
D) $5 x ^ { 2 } - 23 x - 15$
E) $5 x ^ { 2 } + x - 4$

Question 19

Find the value of k such that ​
X - 6 is a factor of x³ - kx² + 2kx - 24.
​

A)​k = 6
B)​k = 10
C)​k = 8
D)​k = 7
E)​k = 9

Question 20

Select the correct graph of the following function.​ $$f ( x ) = 5 x ^ { 3 } - x ^ { 2 } - 12 x + 7$$

A)​
B)​
C)​
D)​
E)​

Question 21

Use the Remainder Theorem and synthetic division to find the function value.Verify your answer using another method.​ $f ( x ) = 4 x ^ { 3 } - 3 x + 8 , f ( - 3 )$ ​

A)94
B)-94
C)-90
D)-91
E)-89

Question 22

Use synthetic division to divide.​ $\left( 3 x ^ { 3 } - 23 x ^ { 2 } + 21 x - 49 \right) \div ( x - 7 )$ ​

A) $2 x ^ { 2 } - 3 x - 7 , x \neq 7$
B) $2 x ^ { 2 } - 3 x + 7 , x \neq 7$
C) $3 x ^ { 2 } - 2 x + 7 , x \neq 7$
D) $3 x ^ { 2 } - 2 x - 7 , x \neq 7$
E) $3 x ^ { 2 } + 2 x + 7 , x \neq 7$

Question 23

Use the Remainder Theorem and synthetic division to find the function value.Verify your answer using another method.​ $f ( x ) = 2 x ^ { 6 } + 8 x ^ { 4 } - 5 x ^ { 2 } + 7 , f ( - 3 )$ ​

A)2,070
B)2,065
C)2,068
D)-2,065
E)2,069

Question 24

Use the Remainder Theorem and synthetic division to find the function value.Verify your answer using another method.​ $f ( x ) = 3 x ^ { 6 } - 4 x ^ { 4 } - 8 x ^ { 2 } + 3 , f ( 4 )$ ​

A)11,141
B)11,140
C)11,139
D)11,136
E)-11,136

Question 25

Simplify the rational expression, $\frac { x ^ { 4 } + 3 x ^ { 3 } - 25 x ^ { 2 } - 39 x + 180 } { x ^ { 2 } - 6 x + 9 }$ ,by using long division or synthetic division.

A) $x ^ { 2 } + 9 x + 20$
B) $x ^ { 2 } + x + 25$
C) $- x ^ { 2 } + 8 x + 25$
D) $$x ^ { 2 } + x - 12$$
E) $- x ^ { 2 } - x + 20$

Question 26

Use long division to divide.​ $8 x ^ { 2 } - 67 x - 45 \div x - 9$ ​

A)​ $8 x ^ { 2 } + 5 , x \neq 9$
B)​ $8 x - 5 , x \neq 9$
C)​ $8 x ^ { 2 } - 5 , x \neq 9$
D)​ $8 x + 5 , x \neq 9$
E)​ $8 x + 9 , x \neq 9$

Question 27

Using the factors (x - 4)and (x + 2),find the remaining factor(s)of $f ( x ) = x ^ { 3 } + x ^ { 2 } - 14 x - 24$ and write the polynomial in fully factored form.

A) $f ( x ) = ( x - 4 ) ( x + 2 ) ^ { 2 }$
B) $f ( x ) = ( x - 4 ) ( x + 2 ) ( x + 3 )$
C) $f ( x ) = ( x - 4 ) ( x + 2 ) ( x + 7 )$
D) $f ( x ) = ( x - 4 ) ^ { 2 } ( x + 2 )$
E) $f ( x ) = ( x - 4 ) ( x + 2 ) ( x - 3 )$

Question 28

If $x = \sqrt { 6 }$ is a root of $x ^ { 3 } + 5 x ^ { 2 } - 6 x - 30 = 0$ ,use synthetic division to factor the polynomial completely and list all real solutions of the equation.

A) $( x + 5 ) ( x - \sqrt { 6 } ) ( x + \sqrt { 6 } ) ; - 5 , \sqrt { 6 } , - \sqrt { 6 }$
B) $( x - 5 ) ( x + \sqrt { 6 } ) ^ { 2 } ; 5 , \sqrt { 6 }$
C) $( x - 5 ) ( x - \sqrt { 6 } ) ^ { 2 } ; 5 , \sqrt { 6 }$
D) $( x - 5 ) ( x + \sqrt { 6 } ) ^ { 2 } ; - 5 , - \sqrt { 6 }$
E) $( x - 5 ) ^ { 2 } ( x - \sqrt { 6 } ) ^ { 2 } ; 5 , \sqrt { 6 }$

Question 29

Use the zero or root feature of a graphing utility to approximate the zeros of $f ( x ) = 2 x ^ { 3 } + 8 x ^ { 2 } - 5 x - 7$ accurate to the nearest thousandth.

A)x = -8.492,-0.695,1.186
B)x = -4.692
C)x = -3.575
D)​x = -3.910,-0.763,1.173
E)​x = -4.388,-0.720,1.108

Question 30

Simplify the rational expression, $\frac { 4 x ^ { 3 } + x ^ { 2 } - 51 x + 36 } { 4 x - 3 }$ ,by using long division or synthetic division.

A) $x ^ { 2 } + 7 x + 16$
B) $x ^ { 2 } - 7 x - 12$
C) $x ^ { 2 } - 6 x + 12$
D) $x ^ { 2 } + 8 x + 9$
E) $x ^ { 2 } + x - 12$

Question 31

Use the Remainder Theorem and synthetic division to find the function value.Verify your answer using another method.​ $f ( x ) = 4 x ^ { 3 } - 3 x + 3 , f \left( \frac { 1 } { 2 } \right)$ ​

A)3
B)4
C)2
D)-1
E)0

Question 32

Write the function in the form f(x)= (x - k)q(x)+ r for the given value of k and demonstrate that f(k)= r.​ $f ( x ) = x ^ { 3 } - 1 x ^ { 2 } - 14 x + 12 \quad k = 4$

A) $( x - 4 ) \left( x ^ { 2 } - 3 x - 2 \right) + 4$
B) $( x + 4 ) \left( x ^ { 2 } + 3 x - 2 \right) + 4$
C) $( x - 4 ) \left( x ^ { 2 } + 3 x - 2 \right) + 4$
D) $( x - 4 ) \left( x ^ { 2 } + 3 x - 2 \right) - 4$
E) $$( x - 4 ) \left( x ^ { 2 } - 3 x - 2 \right) + 4$$

Question 33

Use synthetic division to divide.​ $\left( 5 x ^ { 3 } + 27 x ^ { 2 } + 5 x - 25 \right) \div ( x + 5 )$ ​

A) $5 x ^ { 2 } + 2 x + 5 , x \neq - 5$
B) $5 x ^ { 2 } + 2 x - 5 , x \neq - 5$
C) $5 x ^ { 2 } - 2 x - 5 , x \neq - 5$
D) $2 x ^ { 2 } - 5 x + 5 , x \neq - 5$
E) $2 x ^ { 2 } - 5 x - 5 , x \neq - 5$

Question 34

Use the Remainder Theorem and synthetic division to find the function value.Verify your answer using another method.​ $h ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 9 x + 7 , \quad h ( 2 )$ ​

A)-18
B)-17
C)-21
D)-15
E)-19

Question 35

Use long division to divide.​ $\left( x ^ { 3 } + 343 \right) \div ( x + 7 )$ ​

A) $$- x ^ { 2 } + 7 x + 49 , x \neq - 7$$
B) $x ^ { 2 } - 343 x + 49 , x \neq - 7$
C) $x ^ { 2 } - 7 x - 49 , x \neq - 7$
D) $x ^ { 2 } - 7 x + 49 , x \neq - 7$
E) $x ^ { 2 } + 7 x - 49 , x \neq - 7$

Question 36

Use synthetic division to divide.​ $\left( 4 x ^ { 3 } - 8 x + 24 x ^ { 2 } - 48 \right) \div ( x + 6 )$ ​

A) $4 x ^ { 2 } - 8 , x \neq - 6$
B) $4 x ^ { 2 } + 8 , x \neq - 6$
C) $- 4 x ^ { 2 } - 8 , x \neq - 6$
D) $4 x ^ { 3 } - 8 , x \neq - 6$
E) $4 x ^ { 3 } + 8 , x \neq - 6$

Question 37

If x = $\frac { 2 } { 7 }$ is a root of $49 x ^ { 3 } - 126 x ^ { 2 } + 60 x - 8 = 0$ ,use synthetic division to factor the polynomial completely and list all real solutions of the equation.

A)​ $( 7 x + 2 ) ^ { 2 } ( x - 2 )$ ;x = $- \frac { 2 } { 7 }$ ,2
B) $( 7 x - 2 ) ( x - 2 ) ^ { 2 }$ ;x = $\frac { 2 } { 7 }$ ,2
C) $( 7 x + 2 ) ( x - 2 ) ^ { 2 }$ ;x = $- \frac { 2 } { 7 }$ ,2
D) $( 7 x - 2 ) ^ { 2 } ( x - 2 )$ ;x = $\frac { 2 } { 7 }$ ,2
E) $( 7 x - 2 ) ( 7 x + 2 ) ( x - 2 )$ ;x = $\frac { 2 } { 7 }$ , $- \frac { 2 } { 7 }$ ,2

Question 38

Using the factors (5x + 4)and (x - 1),find the remaining factor(s)of $f ( x ) = - 20 x ^ { 4 } + 109 x ^ { 3 } - 30 x ^ { 2 } - 79 x + 20$ and write the polynomial in fully factored form.

A)​ $f ( x ) = ( 5 x + 4 ) ( - x + 5 ) ( 4 x - 1 ) ( x - 1 )$
B)​ $f ( x ) = ( 5 x + 4 ) ^ { 2 } ( 4 x - 1 ) ( x + 1 )$
C)​ $f ( x ) = ( 5 x + 4 ) ( 5 x + 4 ) ( 4 x - 1 ) ( x - 1 )$
D)​ $f ( x ) = ( 5 x + 4 ) ^ { 2 } ( x - 1 ) ^ { 2 }$
E)​ $f ( x ) = ( 5 x + 4 ) ( - x - 5 ) ^ { 2 } ( x + 1 )$

Question 39

Use long division to divide.​ $2 x ^ { 2 } + 14 x + 20 \div x + 5$ ​

A) $2 x - 4 , x \neq - 5$
B) $2 x + 4 , x \neq - 5$
C) $2 x ^ { 2 } - 4 , x \neq - 5$
D) $2 x ^ { 2 } + 4 , x \neq - 5$
E) $2 x ^ { 2 } + 4 x , x \neq - 5$

Question 40

Use the Remainder Theorem and synthetic division to find the function value.Verify your answer using another method.​ $f ( x ) = 5 x ^ { 3 } - 9 x + 8 , f ( 2 )$ ​

A)-30
B)30
C)32
D)27
E)31

Question 41

Use synthetic division to divide. $\left( - 12 + 2 x ^ { 3 } + 22 x - 12 x ^ { 2 } \right) \div ( x - 3 )$

A)​ $2 x ^ { 2 } - 5 x + 3$
B)​ $2 x ^ { 2 } - 10 x - 8$
C)​ $2 x ^ { 2 } - x + 12$
D)​ $2 x ^ { 2 } - 6 x + 4$
E)​ $2 x ^ { 2 } + 6 x - 2$

Question 42

Use the Remainder Theorem and synthetic division to find each function value.Verify your answers using another method.​ $h ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 6 x + 5 , h ( - 10 )$ ​

A)-1,237
B)-1,231
C)-1,234
D)-1,235
E)-1,233

Question 43

Use synthetic division to perform the division.​ $\frac { 3 x ^ { 3 } - 6 x ^ { 2 } - 23 x + 8 } { x + 2 }$ ​

A) $3 x ^ { 2 } - 12 x + 1 + \frac { 6 } { x + 2 }$
B) $3 x ^ { 2 } - 12 x + 6 + \frac { 8 } { x + 2 }$
C) $3 x ^ { 2 } - 12 x + 1$
D) $x ^ { 2 } - 12 x + 3 - \frac { 6 } { x + 2 }$
E)none of these

Question 44

Use long division to divide.​ $\left( 4 x ^ { 3 } - 21 x ^ { 2 } - 33 x + 28 \right) \div ( 4 x + 7 )$ ​

A) $x ^ { 2 } - 7 x - 4 , x \neq - \frac { 7 } { 4 }$
B) $4 x ^ { 3 } - 7 x + 4 , x \neq - \frac { 7 } { 4 }$
C) $x ^ { 2 } + 7 x + 4 , x \neq - \frac { 7 } { 4 }$
D) $x ^ { 2 } - 7 x + 4 , x \neq - \frac { 7 } { 4 }$
E) $4 x ^ { 2 } - 7 x - 4 , x \neq - \frac { 7 } { 4 }$

Question 45

If x = -2 is a root of $x ^ { 3 } + 5 x ^ { 2 } - 4 x - 20 = 0$ ,use synthetic division to factor the polynomial completely and list all real solutions of the equation.

A) $( x - 5 ) ( x + 2 ) ( x - 2 ) ; x = 5 , - 2,2$
B) $( x + 5 ) ( x + 2 ) ( x - 2 ) ; x = - 5 , - 2,2$
C) $( x + 5 ) ( x + 2 ) ^ { 2 } ; x = - 5 , - 2$
D) $( x + 5 ) ^ { 2 } ( x + 2 ) ; x = - 5 , - 2$
E) $( x + 5 ) ( x - 5 ) ( x - 2 ) ; x = - 5,5,2$

Question 46

Let $$P ( x ) = 7 x ^ { 3 } + 4 x ^ { 2 } - 9 x + 10$$ . ​
Use synthetic division to find the value P (-6).
​

A)-2,608
B)-1,303
C)-1,305
D)-1,304
E)none of these

Question 47

Use synthetic division to divide.​ $\frac { 4 x ^ { 3 } + 20 x ^ { 2 } - 27 x - 18 } { x + \frac { 1 } { 2 } }$ ​

A) $4 x ^ { 2 } + 18 x + 36 , x \neq - \frac { 1 } { 2 }$
B) $4 x ^ { 2 } + 18 x - 36 , x \neq - \frac { 1 } { 2 }$
C)​ $4 x ^ { 2 } - 18 x - 36 , x \neq - \frac { 1 } { 2 }$
D) $- 4 x ^ { 2 } + 18 x - 36 , x \neq - \frac { 1 } { 2 }$
E) $- x ^ { 2 } - 18 x - 36 , x \neq - \frac { 1 } { 2 }$

Question 48

Use long division to divide.​ $\left( x ^ { 3 } - 343 \right) \div ( x - 7 )$ ​

A)​ $x ^ { 2 } + 7 x + 49 , x \neq 7$
B) $x ^ { 2 } - 7 x + 49 , x \neq 7$
C) $x ^ { 2 } + 7 x - 49 , x \neq 7$
D) $x ^ { 2 } - 7 x - 49 , x \neq 7$
E) $- x ^ { 2 } + 7 x + 49 , x \neq 7$

Question 49

Use long division to divide.​ $\left( 6 x ^ { 3 } - 27 x ^ { 2 } + 39 x - 18 \right) \div ( 3 x - 6 )$ ​

A) $2 x ^ { 2 } - 5 x + 3 , x \neq \frac { 6 } { 3 }$
B) $2 x ^ { 2 } - 5 x - 3 , x \neq \frac { 6 } { 3 }$
C) $- 2 x ^ { 2 } - 5 x + 3 , x \neq \frac { 6 } { 3 }$
D) $2 x ^ { 2 } + 5 x + 3 , x \neq \frac { 6 } { 3 }$
E) $2 x ^ { 2 } + 5 x - 3 , x \neq \frac { 6 } { 3 }$

Question 50

Using the factors (x - 5)and (x + 2),find the remaining factor(s)of $f ( x ) = x ^ { 3 } - 6 x ^ { 2 } - x + 30$ and write the polynomial in fully factored form.

A) $f ( x ) = ( x - 5 ) ( x + 2 ) ( x - 3 )$
B) $f ( x ) = ( x - 5 ) ( x + 2 ) ^ { 2 }$
C) $f ( x ) = ( x - 5 ) ( x + 2 ) ( x + 3 )$
D) $f ( x ) = ( x - 5 ) ^ { 2 } ( x + 2 )$
E) $f ( x ) = ( x - 5 ) ( x + 2 ) ( x + 1 )$

Question 51

Use synthetic division to express $P ( x ) = 3 x ^ { 3 } - 10 x ^ { 2 } - 23 x - 35$ in the form (divisor)(quotient)+ remainder for the divisor x - 5.
​

A) $( x - 3 ) \left( 25 x ^ { 2 } + 5 x + 2 \right) - 25$
B) $( x - 5 ) \left( 3 x ^ { 2 } + 5 x + 2 \right) - 25$
C) $( x - 5 ) \left( 3 x ^ { 2 } + 2 \right) - 25$
D) $( x - 5 ) \left( 3 x ^ { 2 } + x + 2 \right)$
E)none of these

Question 52

Use synthetic division to divide.​ $\left( - x ^ { 3 } + 101 x - 825 \right) \div ( x + 11 )$ ​

A) $- x ^ { 2 } + 11 x - 20 , x \neq - 11$
B) $- x ^ { 2 } + 11 x + 20 , x \neq - 11$
C) $x ^ { 2 } + 11 x + 20 , x \neq - 11$
D) $- x ^ { 2 } - 11 x - 20 , x \neq - 11$
E) $x ^ { 2 } + 11 x - 20 , x \neq - 11$

Question 53

Use synthetic division to express $P ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 28 x + 63$ in the form (divisor)(quotient)+ remainder for the divisor x + 5.
​

A) $( x + 5 ) \left( 2 x ^ { 2 } - 7 x + 28 \right) + 7$
B) $( x + 5 ) \left( 2 x ^ { 2 } - 7 x + 7 \right) + 28$
C) $( x - 5 ) \left( 2 x ^ { 2 } - 7 x + 7 \right) - 28$
D) $( x + 5 ) \left( 28 x ^ { 2 } - x + 7 \right) + 28$
E)none of these

Question 54

Write $f ( x ) = x ^ { 3 } + 5 x ^ { 2 } + 15 x + 28$ in the form f (x)= (x - k)q(x)when k = -3.

A) $f ( x ) = ( x + 3 ) \left( x ^ { 2 } + 2 x + 9 \right)$
B) $f ( x ) = ( x + 3 ) \left( x ^ { 2 } + 2 x + 9 \right) + 1$
C) $f ( x ) = ( x - 3 ) \left( x ^ { 2 } + 2 x + 9 \right) - 1$
D) $f ( x ) = ( x - 3 ) \left( x ^ { 2 } - 2 x + 9 \right) + 1$
E) $f ( x ) = ( x + 3 ) \left( x ^ { 2 } - 2 x + 9 \right) - 1$

Question 55

If x = $\frac { 3 } { 5 }$ is a root of $25 x ^ { 3 } - 105 x ^ { 2 } + 99 x - 27 = 0$ ,use synthetic division to factor the polynomial completely and list all real solutions of the equation.

A) $( 5 x - 3 ) ( 5 x + 3 ) ( x - 3 )$ x = $\frac { 3 } { 5 }$ , $- \frac { 3 } { 5 }$ ,3
B) $$( 5 x + 3 ) ^ { 2 } ( x - 3 )$$ x = $- \frac { 3 } { 5 }$ ,3
C)​ $( 5 x - 3 ) ( x - 3 ) ^ { 2 };$ x = $\frac { 3 } { 5 }$ ,3
D) $( 5 x + 3 ) ( x - 3 ) ^ { 2 };$ x = $- \frac { 3 } { 5 }$ ,3
E) $( 5 x - 3 ) ^ { 2 } ( x - 3 );$ x = $\frac { 3 } { 5 }$ ,3

Question 56

Use long division to divide.​ $\left( x ^ { 4 } + 4 x ^ { 2 } + 6 \right) \div \left( x ^ { 2 } + x + 4 \right)$

A) $x ^ { 2 } - x + 1$
B) $x ^ { 2 } + x - 1$
C) $x ^ { 2 } - x + 1 + \frac { 3 x + 2 } { x ^ { 2 } + x + 4 }$
D) $x ^ { 2 } + x - 1 + \frac { 3 x - 4 } { x ^ { 2 } + x + 4 }$
E) $x ^ { 2 } - x + 1 + \frac { 4 } { x ^ { 2 } - x + 1 }$

Question 57

Use synthetic division to divide. $\left( 3 x ^ { 3 } - 19 x ^ { 2 } + 30 x - 8 \right) \div ( x - 4 )$

A)​ $3 x ^ { 2 } - 10 x + 8$
B)​ $3 x ^ { 2 } - 7 x + 2$
C)​ $3 x ^ { 2 } + 7 x + 4$
D)​ $3 x ^ { 2 } - 13 x - 3$
E)​ $3 x ^ { 2 } + 5 x - 6$

Question 58

Use synthetic division to divide.​ $\frac { x ^ { 3 } + 64 } { x + 4 }$ ​

A) $x ^ { 2 } + 4 x + 16 , x \neq - 4$
B) $x ^ { 2 } - 4 x - 16 , x \neq - 4$
C) $x ^ { 2 } + 4 x - 16 , x \neq - 4$
D) $x ^ { 2 } - 4 x + 16 , x \neq - 4$
E)​ $- x ^ { 2 } + 4 x - 16 , x \neq - 4$

Question 59

Use long division to divide.​ $\left( - x ^ { 2 } - 5 x - 6 \right) \div ( x + 2 )$

A) $- x ^ { 2 } - 7 - \frac { 20 } { x + 2 }$
B) $- x - 3$
C) $- x - 7 - \frac { 10 } { x + 2 }$
D)​ $- x - 10$
E)​ $x + 3$

Question 60

Use synthetic division to divide.​ $\frac { x ^ { 3 } - 125 } { x - 5 }$ ​

A)​ $x ^ { 2 } + 5 x + 25 , x \neq 5$
B) $x ^ { 2 } + 5 x - 25 , x \neq - 5$
C) $x ^ { 2 } - 5 x + 25 , x \neq - 5$
D) $- x ^ { 2 } + 5 x + 25 , x \neq - 5$
E) $x ^ { 2 } - 5 x - 25 , x \neq - 5$

Question 61

Find all the rational zeros of the function $f ( x ) = x ^ { 5 } - 5 x ^ { 4 } + 11 x ^ { 3 } - 23 x ^ { 2 } + 28 x - 12$ .

A)​x = -2,2,1,3
B)​x = 3,1
C)​x = -2,-1,-3
D)​x = -1,-2
E)x = -2,2,-1,1,-3

Question 62

Find all rational roots of the equation. ​
147x ³ - 98x ² + 27x - 18 = 0
​

A) $x = \frac { 2 } { 5 }$
B)​ $x = \frac { 2 } { 3 }$
C)​ $x = - \frac { 3 } { 7 } i$
D)​ $x = \frac { 3 } { 7 } i$

Question 63

The number (1 + i)is a root of the equation. ​
X ³ - 10x ² + 18x - 16 = 0
​
Find the other roots of the equation.
​

A)x = 1 - i
B)x = 1 + i
C)x = 5
D)x = 8

Question 64

​Use synthetic division to divide.

Question 65

Use synthetic division to perform the division.​

Question 66

Use synthetic division to perform the division.​ $\frac { 3 x ^ { 4 } - 7 x ^ { 3 } - 16 x + 5 } { x - 3 }$ ​

A) $x ^ { 3 } + x ^ { 2 } + 3 x + 2 - \frac { 11 } { x - 3 }$
B) $3 x ^ { 3 } + 2 x ^ { 2 } + 6 x + 2 + \frac { 11 } { x - 3 }$
C) $3 x ^ { 3 } - 2 x ^ { 2 } + 6 x - 2 + \frac { 11 } { x - 3 }$
D) $3 x ^ { 3 } + 2 x ^ { 2 } + 6 x + 2 + 11$
E)none of these

Question 67

Find all real zeros of the polynomial $f ( x ) = x ^ { 4 } - 113 x ^ { 2 } + 3136$ and determine the multiplicity of each.

A)x = 49,multiplicity 2;x = 64,multiplicity 2
B)x = 7,multiplicity 2;x = 8,multiplicity 2
C)x = 49,multiplicity 2;x = 8,multiplicity 1
D)x = -7,multiplicity 2;x = -8,multiplicity 2
E)x = 7,multiplicity 1;x = -7,multiplicity 1;x = 8,multiplicity 1;x = -8,multiplicity 1

Question 68

Use the zero or root feature of a graphing utility to approximate the zeros of $f ( x ) = 2 x ^ { 3 } + 8 x ^ { 2 } - 5 x - 7$ accurate to the nearest thousandth.

A)​x = -4.692
B)​x = -8.492,-0.695,1.186
C)​x = -3.575
D)​x = -4.388,-0.720,1.108
E)​x = -3.910,-0.763,1.173

Question 69

Use Descartes' rule of signs to find the number of possible positive,negative,and nonreal roots for the following equation. 4x ⁷ + 9x ² + 5x + 10 = 0

A)0 positive;1 negative;6 nonreal
B)2 positive;1 or 3 negative;4 or 0 nonreal
C)0 positive;0 negative;7 nonreal
D)0 positive;0 or 2 negative;5 or 7 nonreal
E)none of these

Question 70

Let .
​
Use synthetic division to find the value P (-7).

Question 71

Find all rational roots of the equation. ​
13x ³ - 2x ² + 52x - 8 = 0
​

A) $x = \frac { 2 } { 3 }$
B)x = -2i
C)x = 2i
D)​ $x = - \frac { 2 } { 13 }$
E)​ $x = \frac { 2 } { 13 }$

Question 72

Find all the real zeros of $f ( x ) = 2 x ^ { 3 } - 6 x ^ { 2 } + 2 x - 6$ .

A)​x = 3, $- \frac { 1 } { 3 }$
B)​x = 2,1,-6
C)​x = ± 1,3
D)​x = 3
E)​x = $\frac { 1 } { 2 }$ , $- \frac { 1 } { 6 }$

Question 73

Find all the rational zeros of the function $f ( x ) = - 2 x ^ { 4 } - x ^ { 3 } + 51 x ^ { 2 } + 25 x - 25$ .

A)x = 1,-5,5,-2
B)x = -2,-5,-5
C)x = -1,-5,5, $\frac { 1 } { 2 }$
D)​x = $- \frac { 1 } { 5 }$ , $\frac { 5 } { 2 }$ , $\frac { 1 } { 2 }$ ,5
E)​x = -2,-5, $\frac { 5 } { 2 }$ , $\frac { 1 } { 2 }$

Question 74

Use Descartes' Rule of Signs to determine the possible number of positive and negative zeros of $f ( x ) = 4 x ^ { 3 } + 3 x ^ { 2 } + 5 x + 1$ .

A)3 positive reals or 1 positive real;3 negative reals or 1 negative real
B)3 positive reals or 1 positive real;no negative reals
C)0 positive real;3 negative reals or 1 negative real
D)1 positive real;3 negative reals or 1 negative real
E)no positive reals;no negative reals

Question 75

Use Descartes' Rule of Signs to determine the possible number of positive and negative zeros of $f ( x ) = 2 x ^ { 3 } - 2 x ^ { 2 } + 3 x - 4$ .

A)3 positive reals or 1 positive real;3 negative reals or 1 negative real
B)3 positive reals or 1 positive real;no negative reals
C)1 positive real;3 negative reals or 1 negative real
D)3 positive reals or 1 positive real;1 negative real
E)no positive reals;no negative reals

Question 76

Find all the real zeros of $f ( x ) = 4 x ^ { 3 } - 20 x ^ { 2 } + 17 x - 4$ .

A)x = $\frac { 1 } { 2 }$ ,4
B)x = $- \frac { 1 } { 2 }$ , $- \frac { 1 } { 4 }$
C)x = $\frac { 1 } { 4 }$ ,-2
D)x = $\frac { 1 } { 2 }$ , $- \frac { 1 } { 4 }$
E)x = $\frac { 1 } { 2 }$ ,-4