Question 1

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.

Accepted Answer

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 
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 D

Question 2

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.

Accepted Answer

A)  $4.37$ 
B)  $\quad 4.21$ 
C)  $\quad 5.21$ 
D)  $\quad 4.69$ 
E)  $\quad 4.94$ 
A)  $4.37$ 
B)  $\quad 4.21$ 
C)  $\quad 5.21$ 
D)  $\quad 4.69$ 
E)  $\quad 4.94$ A

Question 3

Let $P ( x ) = 3 x ^ { 3 } + 2 x ^ { 2 } - 5 x + 5$.
Use synthetic division to find the value $P ( - 5 )$.

Accepted Answer

A)  $\quad - 590$ 
B)  none of these 
C)  $- 295$ 
D)  $\quad - 296$ 
E)  $\quad - 294$ 
A)  $\quad - 590$ 
B)  none of these 
C)  $- 295$ 
D)  $\quad - 296$ 
E)  $\quad - 294$ C

Question 4

Find the interval in which the equation has at least one real root. $$x ^ { 4 } - 8 x ^ { 2 } + 15 = 0$$

Accepted Answer

A)  $\quad - 1$ and 0 
B)  $- 4$ and $- 3$ 
C)  3 and 4 
D)  2 and 3 
E)  $\quad 0$ and 1 
A)  $\quad - 1$ and 0 
B)  $- 4$ and $- 3$ 
C)  3 and 4 
D)  2 and 3 
E)  $\quad 0$ and 1

Question 5

Find integer bounds for the roots of the following equation. $4 x ^ { 3 } - 9 x ^ { 2 } - 58 x = 0$

Accepted Answer

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

Question 6

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 $x - 6$.

Accepted Answer

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 
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 7

Write a third-degree polynomial equation with real coefficients and the roots $8 - i , - 2$.

Accepted Answer

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)  $\quad x ^ { 3 } + 33 x ^ { 2 } + 16 x - 14 = 0$ 
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 8

$$\text { Write a third-degree polynomial equation with real coefficients and the roots } 4 , - i \text {. }$$

Accepted Answer

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 
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 9

Find all rational roots of the equation. $28 x ^ { 4 } + 164 x ^ { 3 } - 213 x ^ { 2 } + 76 x - 7 = 0$

Accepted Answer

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)  $\quad x = \frac { 1 } { 2 } , x = \frac { 1 } { 2 } , x = 9 , x = \frac { 1 } { 7 }$ 
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)  $\quad x = \frac { 1 } { 2 } , x = \frac { 1 } { 2 } , x = 9 , x = \frac { 1 } { 7 }$

Question 10

Tell how many roots the following equation has. $$x ^ { 10 } = 9$$

Accepted Answer

A)  13 
B)  10 
C)  11 
D)  9 
E)  none of these 
A)  13 
B)  10 
C)  11 
D)  9 
E)  none of these

Question 11

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

Accepted Answer

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)  $\quad x ^ { 3 } - 8 x ^ { 2 } + 17$ 
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 12

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 $x - 5$.

Accepted Answer

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 
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 13

Find the negative root of the equation using the bisection method. $x ^ { 2 } - 3 = 0$

Accepted Answer

A)  $- 1.9$ 
B)  $- 1.6$ 
C)  $- 2$ 
D)  $- 1.7$ 
E)  $- 1.5$ 
A)  $- 1.9$ 
B)  $- 1.6$ 
C)  $- 2$ 
D)  $- 1.7$ 
E)  $- 1.5$

Question 14

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.

Accepted Answer

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)  $2,2 , \sqrt { 5 } , - \sqrt { 5 }$ 
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 15

Find the interval in which the equation has at least one real root. $$x ^ { 4 } - 8 x ^ { 2 } + 15 = 0$$

Accepted Answer

A)  $\quad - 1$ and 0 
B)  0 and 1 
C)  $\quad 3$ and 4 
D)  2 and 3 
E)  $\quad - 4$ and $- 3$ 
A)  $\quad - 1$ and 0 
B)  0 and 1 
C)  $\quad 3$ and 4 
D)  2 and 3 
E)  $\quad - 4$ and $- 3$

Question 16

Find the negative root of the equation using the bisection method. $$x ^ { 2 } - 8 = 0$$

Accepted Answer

A)  $- 3$ 
B)  $- 2.8$ 
C)  $- 2.7$ 
D)  $- 2.6$ 
E)  $- 2.5$ 
A)  $- 3$ 
B)  $- 2.8$ 
C)  $- 2.7$ 
D)  $- 2.6$ 
E)  $- 2.5$

Question 17

Write a third-degree polynomial equation with real coefficients and the roots $6 , - i$.

Accepted Answer

A)  none of these 
B)  $\quad x ^ { 3 } - 6 = 0$ 
C)  $\quad x ^ { 3 } - 6 x ^ { 2 } + x - 6 = 0$ 
D)  $\quad x ^ { 3 } - 7 x ^ { 2 } - x + 36 = 0$ 
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 18

Find integer bounds for the roots of the following equation.

Accepted Answer

A) −2,5 
B) −2,6 
C) −3,4 
D)  none of these 
E) −1,4
​	
 
​ 
A) −2,5 
B) −2,6 
C) −3,4 
D)  none of these 
E) −1,4
​	
 
​

Question 19

Find a polynomial with the given zeros. $0,0,0 , \sqrt { 6 } , - \sqrt { 6 }$

Accepted Answer

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 
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 20

Tell how many roots the following equation has. $$x ^ { 7 } = 7$$

Accepted Answer

A)  none of these 
B)  7 
C)  10 
D)  6 
A)  none of these 
B)  7 
C)  10 
D)  6

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.

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.

Let $P ( x ) = 3 x ^ { 3 } + 2 x ^ { 2 } - 5 x + 5$ . Use synthetic division to find the value $P ( - 5 )$ .

Find the interval in which the equation has at least one real root. $x ^ { 4 } - 8 x ^ { 2 } + 15 = 0$

Find integer bounds for the roots of the following equation. $4 x ^ { 3 } - 9 x ^ { 2 } - 58 x = 0$

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 $x - 6$ .

Write a third-degree polynomial equation with real coefficients and the roots $8 - i , - 2$ .

$\text { Write a third-degree polynomial equation with real coefficients and the roots } 4 , - i \text {. }$

Find all rational roots of the equation. $28 x ^ { 4 } + 164 x ^ { 3 } - 213 x ^ { 2 } + 76 x - 7 = 0$

Tell how many roots the following equation has. $x ^ { 10 } = 9$

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

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 $x - 5$ .

Find the negative root of the equation using the bisection method. $x ^ { 2 } - 3 = 0$

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.

Find the interval in which the equation has at least one real root. $x ^ { 4 } - 8 x ^ { 2 } + 15 = 0$

Find the negative root of the equation using the bisection method. $x ^ { 2 } - 8 = 0$

Write a third-degree polynomial equation with real coefficients and the roots $6 , - i$ .

Find integer bounds for the roots of the following equation.

Find a polynomial with the given zeros. $0,0,0 , \sqrt { 6 } , - \sqrt { 6 }$

Tell how many roots the following equation has. $x ^ { 7 } = 7$

A Review of Basic Algebra

Equations and Inequalities

The Rectangular Coordinate System and Graphs of Equations

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Exponential and Logarithmic Functions

Linear Systems

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Exam 6: Solving Polynomial Equations

A partial solution set is given for the equation. x4−6x3−5x2+84x−126=0;{3,3}x ^ { 4 } - 6 x ^ { 3 } - 5 x ^ { 2 } + 84 x - 126 = 0 ; \{ 3,3 \}x4−6x3−5x2+84x−126=0;{3,3} Find the complete solution set.

The profit function for a certain company making and selling xxx units of a product is given by P(x)=−x2+16.37x+52.44P ( x ) = - x ^ { 2 } + 16.37 x + 52.44P(x)=−x2+16.37x+52.44 . Find the smallest value of xxx at which the company breaks even.

Let P(x)=3x3+2x2−5x+5P ( x ) = 3 x ^ { 3 } + 2 x ^ { 2 } - 5 x + 5P(x)=3x3+2x2−5x+5 . Use synthetic division to find the value P(−5)P ( - 5 )P(−5) .

Find the interval in which the equation has at least one real root. x4−8x2+15=0x ^ { 4 } - 8 x ^ { 2 } + 15 = 0x4−8x2+15=0

Find integer bounds for the roots of the following equation. 4x3−9x2−58x=04 x ^ { 3 } - 9 x ^ { 2 } - 58 x = 04x3−9x2−58x=0

Use synthetic division to express P(x)=3x3−16x2−7x−55P ( x ) = 3 x ^ { 3 } - 16 x ^ { 2 } - 7 x - 55P(x)=3x3−16x2−7x−55 in the form ((( divisor) ((( quotient )+) +)+ remainder for the divisor x−6x - 6x−6 .

Write a third-degree polynomial equation with real coefficients and the roots 8−i,−28 - i , - 28−i,−2 .

Write a third-degree polynomial equation with real coefficients and the roots 4,−i. \text { Write a third-degree polynomial equation with real coefficients and the roots } 4 , - i \text {. } Write a third-degree polynomial equation with real coefficients and the roots 4,−i.

Find all rational roots of the equation. 28x4+164x3−213x2+76x−7=028 x ^ { 4 } + 164 x ^ { 3 } - 213 x ^ { 2 } + 76 x - 7 = 028x4+164x3−213x2+76x−7=0

Tell how many roots the following equation has. x10=9x ^ { 10 } = 9x10=9

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

Use synthetic division to express P(x)=5x3−23x2−9x−22P ( x ) = 5 x ^ { 3 } - 23 x ^ { 2 } - 9 x - 22P(x)=5x3−23x2−9x−22 in the form (divisor) (quotient) + remainder for the divisor x−5x - 5x−5 .

Find the negative root of the equation using the bisection method. x2−3=0x ^ { 2 } - 3 = 0x2−3=0

A partial solution set is given for the equation. x4−4x3−6x2+40x−40=0;{2,2}x ^ { 4 } - 4 x ^ { 3 } - 6 x ^ { 2 } + 40 x - 40 = 0 ; \{ 2,2 \}x4−4x3−6x2+40x−40=0;{2,2} Find the complete solution set.

Find the interval in which the equation has at least one real root. x4−8x2+15=0x ^ { 4 } - 8 x ^ { 2 } + 15 = 0x4−8x2+15=0

Find the negative root of the equation using the bisection method. x2−8=0x ^ { 2 } - 8 = 0x2−8=0

Write a third-degree polynomial equation with real coefficients and the roots 6,−i6 , - i6,−i .

Find integer bounds for the roots of the following equation.

Find a polynomial with the given zeros. 0,0,0,6,−60,0,0 , \sqrt { 6 } , - \sqrt { 6 }0,0,0,6​,−6​

Tell how many roots the following equation has. x7=7x ^ { 7 } = 7x7=7

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

Filters

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.

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.

Let $P ( x ) = 3 x ^ { 3 } + 2 x ^ { 2 } - 5 x + 5$ . Use synthetic division to find the value $P ( - 5 )$ .

Find the interval in which the equation has at least one real root. $x ^ { 4 } - 8 x ^ { 2 } + 15 = 0$

Find integer bounds for the roots of the following equation. $4 x ^ { 3 } - 9 x ^ { 2 } - 58 x = 0$

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 $x - 6$ .

Write a third-degree polynomial equation with real coefficients and the roots $8 - i , - 2$ .

$\text { Write a third-degree polynomial equation with real coefficients and the roots } 4 , - i \text {. }$

Find all rational roots of the equation. $28 x ^ { 4 } + 164 x ^ { 3 } - 213 x ^ { 2 } + 76 x - 7 = 0$

Tell how many roots the following equation has. $x ^ { 10 } = 9$

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

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 $x - 5$ .

Find the negative root of the equation using the bisection method. $x ^ { 2 } - 3 = 0$

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.

Find the interval in which the equation has at least one real root. $x ^ { 4 } - 8 x ^ { 2 } + 15 = 0$

Find the negative root of the equation using the bisection method. $x ^ { 2 } - 8 = 0$

Write a third-degree polynomial equation with real coefficients and the roots $6 , - i$ .

Find a polynomial with the given zeros. $0,0,0 , \sqrt { 6 } , - \sqrt { 6 }$

Tell how many roots the following equation has. $x ^ { 7 } = 7$