Question 1

Find integer bounds for the roots of the following equation. $8 x ^ { 3 } - 17 x ^ { 2 } - 178 x = 0$

Accepted Answer

A)  $- 4,4$ 
B)  $- 4,6$ 
C)  $\quad - 5,3$ 
D)  $- 5,5$ 
E)  none of these 
A)  $- 4,4$ 
B)  $- 4,6$ 
C)  $\quad - 5,3$ 
D)  $- 5,5$ 
E)  none of these

Question 2

One root of the equation $x \left( 4 x ^ { 8 } - 3 \right) = 9 x$ is 0 . How many other roots are there?

Accepted Answer

A)  0 
B)  7 
C)  8 
D)  9 
E)  none of these 
A)  0 
B)  7 
C)  8 
D)  9 
E)  none of these

Question 3

Find a polynomial with the given zeros. $7,2,2$

Accepted Answer

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 
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

Question 4

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.

Accepted Answer

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 
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 5

Use Descartes' rule of signs to find the number of possible positive, negative, and nonreal roots for the following equation. $$7 x ^ { 3 } + 5 x ^ { 2 } - 8 x + 8 = 0$$

Accepted Answer

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 
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 6

Let $P ( x ) = x ^ { 4 } - 10 x ^ { 3 } + 6 x + 5 x ^ { 2 } - 4$
Write the terms of $P ( x )$ in descending powers of $x$.

Accepted Answer

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 
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

Question 7

Let $P ( x ) = x ^ { 4 } - 4 x ^ { 3 } + 2 x + 7 x ^ { 2 } - 2$.
Write the terms of $P ( x )$ in descending powers of $x$.

Accepted Answer

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)  $\quad x ^ { 4 } - 4 x ^ { 3 } + 7 x ^ { 2 } + 2 x - 2$ 
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 8

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.

Accepted Answer

A)  $\quad 4.63$ 
B)  $\quad 4.15$ 
C)  $\quad 4.31$ 
D)  $\quad 4.88$ 
E)  $\quad 5.15$ 
A)  $\quad 4.63$ 
B)  $\quad 4.15$ 
C)  $\quad 4.31$ 
D)  $\quad 4.88$ 
E)  $\quad 5.15$

Question 9

Find a polynomial with the given zeros. $1,3,6$

Accepted Answer

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 
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 10

Use synthetic division to perform the division. $$\frac { 3 x ^ { 5 } - 243 x } { x - 3 }$$

Accepted Answer

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)  $\quad 3 x ^ { 4 } + 81$ 
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 11

Find a polynomial with the given zeros. $0,4 - i , 4 + i$

Accepted Answer

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 
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 12

Find the positive solution of the equation using the bisection method. $$x ^ { 2 } = - 1.24 x + 7.344$$

Accepted Answer

A)  $\quad 1.85$ 
B)  $1.24$ 
C)  $\quad 1.69$ 
D)  $\quad 2.02$ 
E)  $\quad 2.16$ 
A)  $\quad 1.85$ 
B)  $1.24$ 
C)  $\quad 1.69$ 
D)  $\quad 2.02$ 
E)  $\quad 2.16$

Question 13

One root of the equation $x \left( 3 x ^ { 3 } - 2 \right) = 7 x$ is 0 . How many other roots are there?

Accepted Answer

A)  4 
B)  none of these 
C)  3
$\begin{array} { l l } \text { d. } & 2 \ \text { e. } & 0 \end{array}$ 
A)  4 
B)  none of these 
C)  3
$\begin{array} { l l } \text { d. } & 2 \ \text { e. } & 0 \end{array}$

Question 14

Use synthetic division to perform the division. $\frac { 3 x ^ { 3 } - 3 x ^ { 2 } - 124 x + 23 } { x + 6 }$

Accepted Answer

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 
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 15

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.

Accepted Answer

A)  $\quad 3,6 , - i , i$ 
B)  $3,3,3 , i$ 
C)  none of these 
D)  $3,6,0,0$ 
E)  $3,6 , - 6 i , 6 i$ 
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 16

Write a fourth-degree polynomial equation with real coefficients and the roots $i , 8 - i$.

Accepted Answer

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)  $\quad x ^ { 4 } - 16 x ^ { 3 } - 65 x ^ { 2 } + 70 x + 16 = 0$ 
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 17

Find the positive solution of the equation using the bisection method. $$x ^ { 2 } = - 1.31 x + 7.665$$

Accepted Answer

A)  $\quad 1.27$ 
B)  $2.19$ 
C)  $1.88$ 
D)  $\quad 2.05$ 
E)  $\quad 1.72$ 
A)  $\quad 1.27$ 
B)  $2.19$ 
C)  $1.88$ 
D)  $\quad 2.05$ 
E)  $\quad 1.72$

Question 18

Find integer bounds for the roots of the following equation. $3 x ^ { 5 } - 11 x ^ { 4 } - 2 x ^ { 3 } + 38 x ^ { 2 } - 21 x - 15 = 0$

Accepted Answer

A)  $- 3,6$ 
B)  $- 2,4$ 
C)  $- 3,3$ 
D)  0,3 
E)  none of these 
A)  $- 3,6$ 
B)  $- 2,4$ 
C)  $- 3,3$ 
D)  0,3 
E)  none of these

Question 19

Use synthetic division to perform the division. $$\frac { 5 x ^ { 5 } - 6,480 x } { x - 6 }$$

Accepted Answer

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 
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 20

Find the interval at which the equation has at least one real root. $3 x ^ { 3 } - 11 x ^ { 2 } - 17 x = 0$

Accepted Answer

A) 2 and 3 
B) 1 and 2 
C) −4 and −5 
D) 4 and 5 
E) −3 and −2
​ 
A) 2 and 3 
B) 1 and 2 
C) −4 and −5 
D) 4 and 5 
E) −3 and −2
​

Find integer bounds for the roots of the following equation. $8 x ^ { 3 } - 17 x ^ { 2 } - 178 x = 0$

One root of the equation $x \left( 4 x ^ { 8 } - 3 \right) = 9 x$ is 0 . How many other roots are there?

Find a polynomial with the given zeros. $7,2,2$

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.

Use Descartes' rule of signs to find the number of possible positive, negative, and nonreal roots for the following equation. $7 x ^ { 3 } + 5 x ^ { 2 } - 8 x + 8 = 0$

Let $P ( x ) = x ^ { 4 } - 10 x ^ { 3 } + 6 x + 5 x ^ { 2 } - 4$ Write the terms of $P ( x )$ in descending powers of $x$ .

Let $P ( x ) = x ^ { 4 } - 4 x ^ { 3 } + 2 x + 7 x ^ { 2 } - 2$ . Write the terms of $P ( x )$ in descending powers of $x$ .

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.

Find a polynomial with the given zeros. $1,3,6$

Use synthetic division to perform the division. $\frac { 3 x ^ { 5 } - 243 x } { x - 3 }$

Find a polynomial with the given zeros. $0,4 - i , 4 + i$

Find the positive solution of the equation using the bisection method. $x ^ { 2 } = - 1.24 x + 7.344$

One root of the equation $x \left( 3 x ^ { 3 } - 2 \right) = 7 x$ is 0 . How many other roots are there?

Use synthetic division to perform the division. $\frac { 3 x ^ { 3 } - 3 x ^ { 2 } - 124 x + 23 } { x + 6 }$

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.

Write a fourth-degree polynomial equation with real coefficients and the roots $i , 8 - i$ .

Find the positive solution of the equation using the bisection method. $x ^ { 2 } = - 1.31 x + 7.665$

Find integer bounds for the roots of the following equation. $3 x ^ { 5 } - 11 x ^ { 4 } - 2 x ^ { 3 } + 38 x ^ { 2 } - 21 x - 15 = 0$

Use synthetic division to perform the division. $\frac { 5 x ^ { 5 } - 6,480 x } { x - 6 }$

Find the interval at which the equation has at least one real root. $3 x ^ { 3 } - 11 x ^ { 2 } - 17 x = 0$

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Exam 6: Solving Polynomial Equations

Find integer bounds for the roots of the following equation. 8x3−17x2−178x=08 x ^ { 3 } - 17 x ^ { 2 } - 178 x = 08x3−17x2−178x=0

One root of the equation x(4x8−3)=9xx \left( 4 x ^ { 8 } - 3 \right) = 9 xx(4x8−3)=9x is 0 . How many other roots are there?

Find a polynomial with the given zeros. 7,2,27,2,27,2,2

A partial solution set is given for the equation. x4−8x3+13x2−8x+12=0;{6,2}x ^ { 4 } - 8 x ^ { 3 } + 13 x ^ { 2 } - 8 x + 12 = 0 ; \{ 6,2 \}x4−8x3+13x2−8x+12=0;{6,2} Find the complete solution set.

Use Descartes' rule of signs to find the number of possible positive, negative, and nonreal roots for the following equation. 7x3+5x2−8x+8=07 x ^ { 3 } + 5 x ^ { 2 } - 8 x + 8 = 07x3+5x2−8x+8=0

Let P(x)=x4−10x3+6x+5x2−4P ( x ) = x ^ { 4 } - 10 x ^ { 3 } + 6 x + 5 x ^ { 2 } - 4P(x)=x4−10x3+6x+5x2−4 Write the terms of P(x)P ( x )P(x) in descending powers of xxx .

Let P(x)=x4−4x3+2x+7x2−2P ( x ) = x ^ { 4 } - 4 x ^ { 3 } + 2 x + 7 x ^ { 2 } - 2P(x)=x4−4x3+2x+7x2−2 . Write the terms of P(x)P ( x )P(x) in descending powers of xxx .

The profit function for a certain company making and selling xxx units of a product is given by P(x)=−x2+20.31x+68.96P ( x ) = - x ^ { 2 } + 20.31 x + 68.96P(x)=−x2+20.31x+68.96 . Find the smallest value of xxx at which the company breaks even.

Find a polynomial with the given zeros. 1,3,61,3,61,3,6

Use synthetic division to perform the division. 3x5−243xx−3\frac { 3 x ^ { 5 } - 243 x } { x - 3 }x−33x5−243x​

Find a polynomial with the given zeros. 0,4−i,4+i0,4 - i , 4 + i0,4−i,4+i

Find the positive solution of the equation using the bisection method. x2=−1.24x+7.344x ^ { 2 } = - 1.24 x + 7.344x2=−1.24x+7.344

One root of the equation x(3x3−2)=7xx \left( 3 x ^ { 3 } - 2 \right) = 7 xx(3x3−2)=7x is 0 . How many other roots are there?

Use synthetic division to perform the division. 3x3−3x2−124x+23x+6\frac { 3 x ^ { 3 } - 3 x ^ { 2 } - 124 x + 23 } { x + 6 }x+63x3−3x2−124x+23​

A partial solution set is given for the equation. x4−9x3+19x2−9x+18=0;{3,6}x ^ { 4 } - 9 x ^ { 3 } + 19 x ^ { 2 } - 9 x + 18 = 0 ; \quad \{ 3,6 \}x4−9x3+19x2−9x+18=0;{3,6} Find the complete solution set.

Write a fourth-degree polynomial equation with real coefficients and the roots i,8−ii , 8 - ii,8−i .

Find the positive solution of the equation using the bisection method. x2=−1.31x+7.665x ^ { 2 } = - 1.31 x + 7.665x2=−1.31x+7.665

Find integer bounds for the roots of the following equation. 3x5−11x4−2x3+38x2−21x−15=03 x ^ { 5 } - 11 x ^ { 4 } - 2 x ^ { 3 } + 38 x ^ { 2 } - 21 x - 15 = 03x5−11x4−2x3+38x2−21x−15=0

Use synthetic division to perform the division. 5x5−6,480xx−6\frac { 5 x ^ { 5 } - 6,480 x } { x - 6 }x−65x5−6,480x​

Find the interval at which the equation has at least one real root. 3x3−11x2−17x=03 x ^ { 3 } - 11 x ^ { 2 } - 17 x = 03x3−11x2−17x=0

A Review of Basic Algebra

Equations and Inequalities

The Rectangular Coordinate System and Graphs of Equations

Functions Functions and Function Notation

Exponential and Logarithmic Functions

Linear Systems

Conic Sections and Quadratic Systems

Natural Number Functions and Probability

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Find integer bounds for the roots of the following equation. $8 x ^ { 3 } - 17 x ^ { 2 } - 178 x = 0$

One root of the equation $x \left( 4 x ^ { 8 } - 3 \right) = 9 x$ is 0 . How many other roots are there?

Find a polynomial with the given zeros. $7,2,2$

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.

Use Descartes' rule of signs to find the number of possible positive, negative, and nonreal roots for the following equation. $7 x ^ { 3 } + 5 x ^ { 2 } - 8 x + 8 = 0$

Let $P ( x ) = x ^ { 4 } - 10 x ^ { 3 } + 6 x + 5 x ^ { 2 } - 4$ Write the terms of $P ( x )$ in descending powers of $x$ .

Let $P ( x ) = x ^ { 4 } - 4 x ^ { 3 } + 2 x + 7 x ^ { 2 } - 2$ . Write the terms of $P ( x )$ in descending powers of $x$ .

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.

Find a polynomial with the given zeros. $1,3,6$

Use synthetic division to perform the division. $\frac { 3 x ^ { 5 } - 243 x } { x - 3 }$

Find a polynomial with the given zeros. $0,4 - i , 4 + i$

Find the positive solution of the equation using the bisection method. $x ^ { 2 } = - 1.24 x + 7.344$

One root of the equation $x \left( 3 x ^ { 3 } - 2 \right) = 7 x$ is 0 . How many other roots are there?

Use synthetic division to perform the division. $\frac { 3 x ^ { 3 } - 3 x ^ { 2 } - 124 x + 23 } { x + 6 }$

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.

Write a fourth-degree polynomial equation with real coefficients and the roots $i , 8 - i$ .

Find the positive solution of the equation using the bisection method. $x ^ { 2 } = - 1.31 x + 7.665$

Find integer bounds for the roots of the following equation. $3 x ^ { 5 } - 11 x ^ { 4 } - 2 x ^ { 3 } + 38 x ^ { 2 } - 21 x - 15 = 0$

Use synthetic division to perform the division. $\frac { 5 x ^ { 5 } - 6,480 x } { x - 6 }$

Find the interval at which the equation has at least one real root. $3 x ^ { 3 } - 11 x ^ { 2 } - 17 x = 0$