Question 1

Write a third-degree polynomial equation with real coefficients and the roots $3 - i , - 1$.

Accepted Answer

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 
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 2

Let $P ( x ) = 7 x ^ { 3 } + 8 x ^ { 2 } - 7 x + 10$.
Use synthetic division to find the value $P ( 2 )$.

Accepted Answer

A)  83 
B)  85 
C)  $\quad 168$ 
D)  none of these 
E)  84 
A)  83 
B)  85 
C)  $\quad 168$ 
D)  none of these 
E)  84

Question 3

Find integer bounds for the roots of the following equation. $$x ^ { 2 } - 2 x - 7 = 0$$

Accepted Answer

A)  $- 2,4$ 
B)  $- 1,3$ 
C)  $- 2,6$ 
D)  $- 3,3$ 
E)  none of these 
A)  $- 2,4$ 
B)  $- 1,3$ 
C)  $- 2,6$ 
D)  $- 3,3$ 
E)  none of these

Question 4

Find a polynomial with the given zeros. $0,0,0 , \sqrt { 5 } , - \sqrt { 5 }$

Accepted Answer

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)  $x ^ { 2 } + 5 x + 7$ 
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 5

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

Question 6

Find the interval at which the equation has at least one real root. $3 x ^ { 3 } - 11 x ^ { 2 } - 15 x = 0$

Accepted Answer

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

Question 7

Use Descartes' rule of signs to find the number of possible positive, negative, and nonreal roots for the following equation. $$3 x ^ { 3 } + 8 x ^ { 2 } - 4 x + 10 = 0$$

Accepted Answer

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 
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 8

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

Accepted Answer

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

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

Accepted Answer

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 
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 10

Use synthetic division to perform the division. $$\frac { 3 x ^ { 3 } - 7 x ^ { 2 } - 194 x + 37 } { x + 7 }$$

Accepted Answer

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)  $x ^ { 2 } - 28 x + 3 - \frac { 23 } { x + 7 }$ 
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)  $x ^ { 2 } - 28 x + 3 - \frac { 23 } { x + 7 }$

Write a third-degree polynomial equation with real coefficients and the roots $3 - i , - 1$ .

Let $P ( x ) = 7 x ^ { 3 } + 8 x ^ { 2 } - 7 x + 10$ . Use synthetic division to find the value $P ( 2 )$ .

Find integer bounds for the roots of the following equation. $x ^ { 2 } - 2 x - 7 = 0$

Find a polynomial with the given zeros. $0,0,0 , \sqrt { 5 } , - \sqrt { 5 }$

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

Find the interval at which the equation has at least one real root. $3 x ^ { 3 } - 11 x ^ { 2 } - 15 x = 0$

Use Descartes' rule of signs to find the number of possible positive, negative, and nonreal roots for the following equation. $3 x ^ { 3 } + 8 x ^ { 2 } - 4 x + 10 = 0$

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

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

Use synthetic division to perform the division. $\frac { 3 x ^ { 3 } - 7 x ^ { 2 } - 194 x + 37 } { x + 7 }$

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

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Exam 6: Solving Polynomial Equations

Write a third-degree polynomial equation with real coefficients and the roots 3−i,−13 - i , - 13−i,−1 .

Let P(x)=7x3+8x2−7x+10P ( x ) = 7 x ^ { 3 } + 8 x ^ { 2 } - 7 x + 10P(x)=7x3+8x2−7x+10 . Use synthetic division to find the value P(2)P ( 2 )P(2) .

Find integer bounds for the roots of the following equation. x2−2x−7=0x ^ { 2 } - 2 x - 7 = 0x2−2x−7=0

Find a polynomial with the given zeros. 0,0,0,5,−50,0,0 , \sqrt { 5 } , - \sqrt { 5 }0,0,0,5​,−5​

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

Find the interval at which the equation has at least one real root. 3x3−11x2−15x=03 x ^ { 3 } - 11 x ^ { 2 } - 15 x = 03x3−11x2−15x=0

Use Descartes' rule of signs to find the number of possible positive, negative, and nonreal roots for the following equation. 3x3+8x2−4x+10=03 x ^ { 3 } + 8 x ^ { 2 } - 4 x + 10 = 03x3+8x2−4x+10=0

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

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

Use synthetic division to perform the division. 3x3−7x2−194x+37x+7\frac { 3 x ^ { 3 } - 7 x ^ { 2 } - 194 x + 37 } { x + 7 }x+73x3−7x2−194x+37​

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

Write a third-degree polynomial equation with real coefficients and the roots $3 - i , - 1$ .

Let $P ( x ) = 7 x ^ { 3 } + 8 x ^ { 2 } - 7 x + 10$ . Use synthetic division to find the value $P ( 2 )$ .

Find integer bounds for the roots of the following equation. $x ^ { 2 } - 2 x - 7 = 0$

Find a polynomial with the given zeros. $0,0,0 , \sqrt { 5 } , - \sqrt { 5 }$

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

Find the interval at which the equation has at least one real root. $3 x ^ { 3 } - 11 x ^ { 2 } - 15 x = 0$

Use Descartes' rule of signs to find the number of possible positive, negative, and nonreal roots for the following equation. $3 x ^ { 3 } + 8 x ^ { 2 } - 4 x + 10 = 0$

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

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

Use synthetic division to perform the division. $\frac { 3 x ^ { 3 } - 7 x ^ { 2 } - 194 x + 37 } { x + 7 }$