Deck 6: Solving Polynomial Equations

Question 1

Tell how many roots the following equation has. $$x ^ { 10 } = 9$$
a. 13
b. 10
c. 11
d. 9
e. none of these

Answer 1

B

Question 2

Let $P ( x ) = x ^ { 4 } - 10 x ^ { 3 } + 6 x + 5 x ^ { 2 } - 4$
Write the terms of $P ( x )$ in descending powers of $x$ .
a. $x ^ { 4 } + 5 x ^ { 2 } - 4$
b. $\quad x ^ { 4 } - 10 x ^ { 3 } + 5 x ^ { 2 } + 6 x - 4$
c. $\quad 4 - 6 x + 5 x ^ { 2 } + 10 x ^ { 3 } - x ^ { 4 }$
d. $\quad 10 x ^ { 3 } + 6 x - 4$
e. none of these

Answer 2

B

Question 3

Find a polynomial with the given zeros. $7,2,2$
a. $\quad x ^ { 3 } - 11 x ^ { 2 } + 32 x - 28$
b. $\quad 32 x ^ { 3 } - 11 x ^ { 2 } + 32 x - 28$
c. $\quad x ^ { 3 } + 11 x ^ { 2 } - 32 x + 28$
d. $\quad 11 x ^ { 2 } + 32 x - 28$
e. none of these

Answer 3

A

Question 4

Use Descartes' rule of signs to find the number of possible positive, negative, and nonreal roots for the following equation. $$3 x ^ { 3 } + 8 x ^ { 2 } - 4 x + 10 = 0$$
a. $\quad 1$ positive; 1 negative; 1 nonreal
b. $\quad 0$ positive; 0 negative; 3 nonreal
c. $\quad$ positive; 0 or 2 negative; 0 or 2 nonreal
d. $\quad 0$ or 2 positive; 1 negative; 0 or 2 nonreal
e. none of these

Question 5

Use synthetic division to perform the division. $\frac { 3 x ^ { 3 } - 3 x ^ { 2 } - 124 x + 23 } { x + 6 }$
a. $\quad 3 x ^ { 2 } - 21 x + 2$
b. $\quad x ^ { 2 } - 21 x + 3 - \frac { 11 } { x + 6 }$
c. $3 x ^ { 2 } - 21 x + 2 + \frac { 11 } { x + 6 }$
d. $\quad 3 x ^ { 2 } - 21 x + 11 + \frac { 23 } { x + 6 }$
e. none of these

Question 6

One root of the equation $x \left( 4 x ^ { 8 } - 3 \right) = 9 x$ is 0 . How many other roots are there?
a. 0
b. 7
c. 8
d. 9
e. none of these

Question 7

A partial solution set is given for the equation. $$x ^ { 4 } - 6 x ^ { 3 } - 5 x ^ { 2 } + 84 x - 126 = 0 ; \{ 3,3 \}$$
Find the complete solution set.
a. $\quad 3,3 , \sqrt { 17 } , - \sqrt { 17 }$
b. $\quad 7,7 , \sqrt { 14 } , - \sqrt { 14 }$
c. $3,3 , \sqrt { 7 } , - \sqrt { 7 }$
d. $\quad 3,3 , \sqrt { 14 } , - \sqrt { 14 }$
e. none of these

Question 8

Find a polynomial with the given zeros. $0,0,0 , \sqrt { 6 } , - \sqrt { 6 }$
a. $x ^ { 5 } - x ^ { 3 } + 6 x$
b. $\quad x ^ { 5 } - 3 x ^ { 3 }$
c. $\quad x ^ { 2 } + 6 x + 3$
d. $x ^ { 5 } - 6 x ^ { 3 }$
e. none of these

Question 9

Let $P ( x ) = 3 x ^ { 3 } + 2 x ^ { 2 } - 5 x + 5$ .
Use synthetic division to find the value $P ( - 5 )$ .
a. $\quad - 590$
b. none of these
c. $- 295$
d. $\quad - 296$
e. $\quad - 294$

Question 10

Write a fourth-degree polynomial equation with real coefficients and the roots $i , 8 - i$ .
a. $\quad x ^ { 4 } + 65 x ^ { 2 } - 16 x - 70 = 0$
b. $\quad x ^ { 4 } - 16 x ^ { 3 } - 65 x ^ { 2 } + 70 x + 16 = 0$
c. $\quad x ^ { 4 } + 66 x ^ { 2 } - 16 x + 65 = 0$
d. $\quad x ^ { 4 } - 16 x ^ { 3 } + 66 x ^ { 2 } - 16 x + 65 = 0$
e. none of these

Question 11

Find integer bounds for the roots of the following equation. $$x ^ { 2 } - 2 x - 7 = 0$$
a. $- 2,4$
b. $- 1,3$
c. $- 2,6$
d. $- 3,3$
e. none of these

Question 12

Find a polynomial with the given zeros. $0,4 - i , 4 + i$
a. $x ^ { 3 } + 17 x ^ { 2 } - 4 x$
b. $x ^ { 3 } - 4 x ^ { 2 } + 16 x$
c. $\quad x ^ { 3 } - 8 x ^ { 2 } + 17$
d. $x ^ { 3 } - 8 x ^ { 2 } + 17 x$
e. none of these

Question 13

Find integer bounds for the roots of the following equation. $3 x ^ { 5 } - 11 x ^ { 4 } - 2 x ^ { 3 } + 38 x ^ { 2 } - 21 x - 15 = 0$
a. $- 3,6$
b. $- 2,4$
c. $- 3,3$
d. 0,3
e. none of these

Question 14

Use synthetic division to express $P ( x ) = 3 x ^ { 3 } - 16 x ^ { 2 } - 7 x - 55$ in the form $($ divisor) $($ quotient $) +$ remainder for the divisor $x - 6$ .
a. $\quad ( x - 6 ) \left( 3 x ^ { 2 } + 5 \right) - 25$
b. $( x - 3 ) \left( 25 x ^ { 2 } + 2 x + 5 \right) - 25$
c. $( x - 6 ) \left( 3 x ^ { 2 } + 2 x + 5 \right) - 25$
d. $( x - 6 ) \left( 3 x ^ { 2 } + x + 5 \right)$
e. none of these

Question 15

Write a third-degree polynomial equation with real coefficients and the roots $3 - i , - 1$ .
a. $\quad x ^ { 3 } + 7 x ^ { 2 } - 10 = 0$
b. $\quad x ^ { 3 } + 4 x ^ { 2 } + 7 x - 5 = 0$
c. $\quad x ^ { 3 } + 10 = 0$
d. $\quad x ^ { 3 } - 5 x ^ { 2 } + 4 x + 10 = 0$
e. none of these

Question 16

$$\text { Write a third-degree polynomial equation with real coefficients and the roots } 4 , - i \text {. }$$ a. $\quad x ^ { 3 } - 4 = 0$
b. $\quad x ^ { 3 } - 4 x ^ { 2 } + x - 4 = 0$
c. $\quad x ^ { 3 } - 16 = 0$
d. $\quad x ^ { 3 } - 5 x ^ { 2 } - x + 16 = 0$
e. none of these

Question 17

Use synthetic division to perform the division. $$\frac { 5 x ^ { 5 } - 6,480 x } { x - 6 }$$
a. $\quad 5 x ^ { 4 } + x ^ { 3 } - 30 x ^ { 2 } + x + 1,080$
b. $\quad 5 x ^ { 4 } - x ^ { 3 } + 30 x ^ { 2 } - 180 x + 1,080$
c. $\quad 5 x ^ { 4 } + 1,080$
d. $5 x ^ { 4 } + 30 x ^ { 3 } + 180 x ^ { 2 } + 1,080 x$
e. none of these

Question 18

Find all rational roots of the equation. $28 x ^ { 4 } + 164 x ^ { 3 } - 213 x ^ { 2 } + 76 x - 7 = 0$
a. $x = \frac { 1 } { 4 } , x = \frac { 1 } { 2 } , x = - 7 , x = \frac { 1 } { 9 }$
b. $\quad x = \frac { 1 } { 4 } , x = \frac { 1 } { 4 } , x = 7 , x = \frac { 1 } { 7 }$
c. $x = \frac { 1 } { 2 } , x = \frac { 1 } { 2 } , x = - 7 , x = \frac { 1 } { 7 }$
d. $x = \frac { 1 } { 2 } , x = \frac { 1 } { 2 } , x = 9 , x = \frac { 1 } { 7 }$

Question 19

Find integer bounds for the roots of the following equation. $8 x ^ { 3 } - 17 x ^ { 2 } - 178 x = 0$
a. $- 4,4$
b. $- 4,6$
c. $\quad - 5,3$
d. $- 5,5$
e. none of these

Question 20

A partial solution set is given for the equation. $$x ^ { 4 } - 8 x ^ { 3 } + 13 x ^ { 2 } - 8 x + 12 = 0 ; \{ 6,2 \}$$
Find the complete solution set.
a. $6,2 , - i , i$
b. $6,2,0,0$
c. $6,6,6 , i$
d. $6,2 , - 2 i , 2 i$
e. none of these

Question 21

Find the positive solution of the equation using the bisection method. $$x ^ { 2 } = - 1.31 x + 7.665$$
a. $\quad 1.27$
b. $2.19$
c. $1.88$
d. $\quad 2.05$
e. $\quad 1.72$

Question 22

Write a third-degree polynomial equation with real coefficients and the roots $6 , - i$ .
a. none of these
b. $\quad x ^ { 3 } - 6 = 0$
c. $\quad x ^ { 3 } - 6 x ^ { 2 } + x - 6 = 0$
d. $\quad x ^ { 3 } - 7 x ^ { 2 } - x + 36 = 0$

Question 23

Find the interval at which the equation has at least one real root. $$3 x ^ { 3 } - 11 x ^ { 2 } - 15 x = 0$$
$\begin{array} { l l } \text { a. } & - 3 \text { and } - 2 \\ \text { b. } & - 4 \text { and } - 5 \\ \text { c. } & 4 \text { and } 5 \\ \text { d. } & 1 \text { and } 2 \\ \text { e. } & 2 \text { and } 3 \end{array}$

Question 24

The profit function for a certain company making and selling $x$ units of a product is given by $P ( x ) = - x ^ { 2 } + 16.37 x + 52.44$ . Find the smallest value of $x$ at which the company breaks even.
a. $4.37$
b. $\quad 4.21$
c. $\quad 5.21$
d. $\quad 4.69$
e. $\quad 4.94$

Question 25

Find a polynomial with the given zeros. $$0,4 - i , 4 + i$$
a. $\quad x ^ { 3 } + 17 x ^ { 2 } - 4 x$
b. $x ^ { 3 } - 4 x ^ { 2 } + 16 x$
c. none of these
d. $\quad x ^ { 3 } - 8 x ^ { 2 } + 17 x$
e. $\quad x ^ { 3 } - 8 x ^ { 2 } + 17$

Question 26

Write a third-degree polynomial equation with real coefficients and the roots $8 - i , - 2$ .
a. $\quad x ^ { 3 } + 16 x ^ { 2 } - 130 = 0$
b. none of these
c. $\quad x ^ { 3 } + 130 = 0$
d. $\quad x ^ { 3 } - 14 x ^ { 2 } + 33 x + 130 = 0$
e. $\quad x ^ { 3 } + 33 x ^ { 2 } + 16 x - 14 = 0$

Question 27

Let $P ( x ) = x ^ { 4 } - 4 x ^ { 3 } + 2 x + 7 x ^ { 2 } - 2$ .
Write the terms of $P ( x )$ in descending powers of $x$ .
a. none of these
b. $\quad x ^ { 4 } + 7 x ^ { 2 } - 2$
c. $\quad 4 x ^ { 3 } + 2 x - 2$
d. $\quad 2 - 2 x + 7 x ^ { 2 } + 4 x ^ { 3 } - x ^ { 4 }$
e. $\quad x ^ { 4 } - 4 x ^ { 3 } + 7 x ^ { 2 } + 2 x - 2$

Question 28

A partial solution set is given for the equation. $$x ^ { 4 } - 4 x ^ { 3 } - 6 x ^ { 2 } + 40 x - 40 = 0 ; \{ 2,2 \}$$
Find the complete solution set.
a. $\quad 2,2 , \sqrt { 10 } , - \sqrt { 10 }$
b. $\quad 5,5 , \sqrt { 10 } , - \sqrt { 10 }$
c. $2,2 , \sqrt { 12 } , - \sqrt { 12 }$
d. none of these
e. $2,2 , \sqrt { 5 } , - \sqrt { 5 }$

Question 29

Write a fourth-degree polynomial equation with real coefficients and the roots $i , 8 - i$ .
a. $\quad x ^ { 4 } - 16 x ^ { 3 } + 66 x ^ { 2 } - 16 x + 65 = 0$
b. $\quad x ^ { 4 } + 66 x ^ { 2 } - 16 x + 65 = 0$
c. none of these
d. $\quad x ^ { 4 } + 65 x ^ { 2 } - 16 x - 70 = 0$
e. $\quad x ^ { 4 } - 16 x ^ { 3 } - 65 x ^ { 2 } + 70 x + 16 = 0$

Question 30

Find the negative root of the equation using the bisection method. $$x ^ { 2 } - 8 = 0$$
a. $- 3$
b. $- 2.8$
c. $- 2.7$
d. $- 2.6$
e. $- 2.5$

Question 31

Use synthetic division to express $P ( x ) = 5 x ^ { 3 } - 23 x ^ { 2 } - 9 x - 22$ in the form (divisor) (quotient) + remainder for the divisor $x - 5$ .
a. $( x - 5 ) \left( 17 x ^ { 2 } + 2 x + 1 \right) - 17$
b. $\quad ( x - 5 ) \left( 5 x ^ { 2 } + 1 \right) - 17$
c. $\quad ( x - 5 ) \left( 5 x ^ { 2 } + x + 1 \right)$
d. $( x - 5 ) \left( 5 x ^ { 2 } + 2 x + 1 \right) - 17$
e. none of these

Question 32

Find a polynomial with the given zeros. $1,3,6$
a. $\quad 10 x ^ { 2 } + 27 x - 18$
b. $\quad x ^ { 3 } - 10 x ^ { 2 } + 27 x - 18$
c. $\quad 27 x ^ { 3 } - 10 x ^ { 2 } + 27 x - 18$
d. $\quad x ^ { 3 } + 10 x ^ { 2 } - 27 x + 18$
e. none of these

Question 33

A partial solution set is given for the equation. $$x ^ { 4 } - 9 x ^ { 3 } + 19 x ^ { 2 } - 9 x + 18 = 0 ; \quad \{ 3,6 \}$$
Find the complete solution set.
a. $\quad 3,6 , - i , i$
b. $3,3,3 , i$
c. none of these
d. $3,6,0,0$
e. $3,6 , - 6 i , 6 i$

Question 34

Find the interval in which the equation has at least one real root. $$x ^ { 4 } - 8 x ^ { 2 } + 15 = 0$$
a. $\quad - 1$ and 0
b. $- 4$ and $- 3$
c. 3 and 4
d. 2 and 3
e. $\quad 0$ and 1

Question 35

Use synthetic division to perform the division. $$\frac { 3 x ^ { 5 } - 243 x } { x - 3 }$$
a. $\quad 3 x ^ { 4 } + 9 x ^ { 3 } + 27 x ^ { 2 } + 81 x$
b. none of these
c. $\quad 3 x ^ { 4 } + x ^ { 3 } - 9 x ^ { 2 } + x + 81$
d. $\quad 3 x ^ { 4 } - x ^ { 3 } + 9 x ^ { 2 } - 27 x + 81$
e. $\quad 3 x ^ { 4 } + 81$

Question 36

Let $P ( x ) = 7 x ^ { 3 } + 8 x ^ { 2 } - 7 x + 10$ .
Use synthetic division to find the value $P ( 2 )$ .
a. 83
b. 85
c. $\quad 168$
d. none of these
e. 84

Question 37

Tell how many roots the following equation has. $$x ^ { 7 } = 7$$
a. none of these
b. 7
c. 10
d. 6

Question 38

Use synthetic division to perform the division. $$\frac { 3 x ^ { 3 } - 7 x ^ { 2 } - 194 x + 37 } { x + 7 }$$
a. none of these
b. $\quad 3 x ^ { 2 } - 28 x + 2$
c. $\quad 3 x ^ { 2 } - 28 x + 23 + \frac { 37 } { x + 7 }$
d. $\quad 3 x ^ { 2 } - 28 x + 2 + \frac { 23 } { x + 7 }$
e. $x ^ { 2 } - 28 x + 3 - \frac { 23 } { x + 7 }$

Question 39

Find a polynomial with the given zeros. $0,0,0 , \sqrt { 5 } , - \sqrt { 5 }$
a. $x ^ { 5 } - 5 x ^ { 3 }$
b. $\quad x ^ { 5 } - x ^ { 3 } + 5 x$
c. $\quad x ^ { 5 } - 7 x ^ { 3 }$
d. none of these
e. $x ^ { 2 } + 5 x + 7$

Question 40

One root of the equation $x \left( 3 x ^ { 3 } - 2 \right) = 7 x$ is 0 . How many other roots are there?
a. 4
b. none of these
c. 3
$\begin{array} { l l } \text { d. } & 2 \\ \text { e. } & 0 \end{array}$

Question 41

The profit function for a certain company making and selling $x$ units of a product is given by $P ( x ) = - x ^ { 2 } + 20.31 x + 68.96$ . Find the smallest value of $x$ at which the company breaks even.
a. $\quad 4.63$
b. $\quad 4.15$
c. $\quad 4.31$
d. $\quad 4.88$
e. $\quad 5.15$

Question 42

Find the positive solution of the equation using the bisection method. $$x ^ { 2 } = - 1.24 x + 7.344$$
a. $\quad 1.85$
b. $1.24$
c. $\quad 1.69$
d. $\quad 2.02$
e. $\quad 2.16$

Question 43

Use Descartes' rule of signs to find the number of possible positive, negative, and nonreal roots for the following equation. $$7 x ^ { 3 } + 5 x ^ { 2 } - 8 x + 8 = 0$$
a. $\quad 0$ or 2 positive; 1 negative; 0 or 2 nonreal
b. $\quad 0$ positive; 0 negative; 3 nonreal
c. $\quad 1$ positive; 0 or 2 negative; 0 or 2 nonreal
d. $\quad$ l positive; 1 negative; 1 nonreal
e. none of these

Question 44

Find the negative root of the equation using the bisection method. $$x ^ { 2 } - 3 = 0$$
a. $- 1.9$
b. $- 1.6$
c. $\quad - 2$
$\begin{array} { l l } \text { d. } & - 1.7 \\ \text { e. } & - 1.5 \end{array}$

Question 45

Find integer bounds for the roots of the following equation. $3 x ^ { 5 } - 11 x ^ { 4 } - 2 x ^ { 3 } + 38 x ^ { 2 } - 21 x - 15 = 0$
a. none of these
b. $- 3,6$
c. $- 2,4$
d. $- 3,3$
e. $\quad 0,3$

Question 46

Find the interval at which the equation has at least one real root. $$3 x ^ { 3 } - 11 x ^ { 2 } - 17 x = 0$$
$\begin{array} { l l } \text { a. } & 2 \text { and } 3 \\ \text { b. } & 1 \text { and } 2 \\ \text { c. } & - 4 \text { and } - 5 \\ \text { d. } & 4 \text { and } 5 \\ \text { e. } & - 3 \text { and } - 2 \end{array}$

Question 47

Find integer bounds for the roots of the following equation. $4 x ^ { 3 } - 9 x ^ { 2 } - 58 x = 0$
a. $- 4,5$
b. $- 4,3$
c. $- 3,6$
d. none of these
e. $- 3,4$

Question 48

Find the interval in which the equation has at least one real root. $$x ^ { 4 } - 8 x ^ { 2 } + 15 = 0$$
a. $\quad - 1$ and 0
b. 0 and 1
c. $\quad 3$ and 4
d. 2 and 3
e. $\quad - 4$ and $- 3$

Question 49

Find integer bounds for the roots of the following equation. $$\begin{array}{l}x ^ { 2 } - 3 x - 7 = 0\\\begin{array} { l l } \text { a. } & - 2,5 \\\text { b. } & - 2,6 \\\text { c. } & - 3,4 \\\text { d. } & \text { none of these } \\\text { e. } & - 1,4\end{array}\end{array}$$

Question 50

Find all rational roots of the equation. $28 x ^ { 4 } + 164 x ^ { 3 } - 213 x ^ { 2 } + 76 x - 7 = 0$
a. $\quad x = \frac { 1 } { 4 } , x = \frac { 1 } { 2 } , x = - 7 , x = \frac { 1 } { 9 }$
b. $\quad x = \frac { 1 } { 4 } , x = \frac { 1 } { 4 } , x = 7 , x = \frac { 1 } { 7 }$
c. $x = \frac { 1 } { 2 } , x = \frac { 1 } { 2 } , x = - 7 , x = \frac { 1 } { 7 }$
d. $\quad x = \frac { 1 } { 2 } , x = \frac { 1 } { 2 } , x = 9 , x = \frac { 1 } { 7 }$