Question 1

Determine upper and lower bounds for the real roots of the equation. 
5x 4  - 10x - 12 = 0

Accepted Answer

A)  2 is an upper bound, - 1 is a lower bound 
B)  8 is an upper bound, - 1 is a lower bound 
C)  1 is an upper bound, - 2 is a lower bound 
D)  2 is an upper bound, - 4 is a lower bound 
E)  8 is an upper bound, - 4 is a lower bound 
A)  2 is an upper bound, - 1 is a lower bound 
B)  8 is an upper bound, - 1 is a lower bound 
C)  1 is an upper bound, - 2 is a lower bound 
D)  2 is an upper bound, - 4 is a lower bound 
E)  8 is an upper bound, - 4 is a lower bound A

Question 2

Determine the constants (denoted with capital letters) so that the equation is an identity. $\frac { 8 x + 3 } { ( x + 3 ) ^ { 2 } } = \frac { A } { x + 3 } + \frac { B } { ( x + 3 ) ^ { 2 } }$

Accepted Answer

A)  $\begin{array} { l } 
A = 7 \
B = 21
\end{array}$ 
B)  $\begin{array} { l } 
A = 7 \
B = 22
\end{array}$ 
C)  $\begin{array} { l } 
A = - 8 \
B = - 21
\end{array}$ 
D)  $\begin{array} { l } 
A = 8 \
B = - 21
\end{array}$ 
E)  $\begin{array} { l } 
A = - 8 \
B = 21
\end{array}$ 
A)  $\begin{array} { l } 
A = 7 \
B = 21
\end{array}$ 
B)  $\begin{array} { l } 
A = 7 \
B = 22
\end{array}$ 
C)  $\begin{array} { l } 
A = - 8 \
B = - 21
\end{array}$ 
D)  $\begin{array} { l } 
A = 8 \
B = - 21
\end{array}$ 
E)  $\begin{array} { l } 
A = - 8 \
B = 21
\end{array}$ D

Question 3

Determine which of the following polynomials is not reducible.

Accepted Answer

A)  $x ^ { 2 } + 18$ 
B)  $x ^ { 2 } - 4$ 
C)  $x ^ { 2 } - 1$ 
D)  $x ^ { 2 } - 21$ 
E)  $x ^ { 2 } - 18$ 
A)  $x ^ { 2 } + 18$ 
B)  $x ^ { 2 } - 4$ 
C)  $x ^ { 2 } - 1$ 
D)  $x ^ { 2 } - 21$ 
E)  $x ^ { 2 } - 18$ A

Question 4

Find a quadratic equation with rational coefficients, one of whose roots is the given number. Write your answer so that the coefficient of 
$x ^ { 2 }$ is 1. $r _ { 1 } = 1 + \sqrt { 2 }$

Accepted Answer

A)  $x ^ { 2 } - 3 x - 1 = 0$ 
B)  $x ^ { 2 } - 2 x - 8 = 0$ 
C)  $x ^ { 2 } - 2 x - 1 = 0$ 
D)  $x ^ { 2 } - 2 x - 9 = 0$ 
E)  $x ^ { 2 } - 2 x - 3 = 0$ 
A)  $x ^ { 2 } - 3 x - 1 = 0$ 
B)  $x ^ { 2 } - 2 x - 8 = 0$ 
C)  $x ^ { 2 } - 2 x - 1 = 0$ 
D)  $x ^ { 2 } - 2 x - 9 = 0$ 
E)  $x ^ { 2 } - 2 x - 3 = 0$

Question 5

Use Descartes's rule of signs to obtain information regarding the roots of the equation. $x ^ { 9 } + 5 = 0$

Accepted Answer

A)  1 negative root, 8 complex roots 
B)  9 complex roots 
C)  1 negative root, 9 complex roots 
D)  1 positive root, 1 negative real root, 7 complex roots 
E)  1 positive root, 8 complex roots 
A)  1 negative root, 8 complex roots 
B)  9 complex roots 
C)  1 negative root, 9 complex roots 
D)  1 positive root, 1 negative real root, 7 complex roots 
E)  1 positive root, 8 complex roots

Question 6

Determine the constants (denoted by capital letters) so that the equation is an identity. $\frac { x - 17 } { x \left( x ^ { 2 } + 2 \right) } = \frac { A } { x } + \frac { B x + C } { x ^ { 2 } + 2 }$

Accepted Answer

A)  $\begin{array} { l } 
A = 8 \frac { 1 } { 2 } \
B = 1 \
C = 8 \frac { 1 } { 2 }
\end{array}$ 
B)  $\begin{array} { l } 
A = - 8 \frac { 1 } { 2 } \
B = 8 \frac { 1 } { 2 } \
C = 1
\end{array}$ 
C)  $\begin{array} { l } 
A = 8 \frac { 1 } { 2 } \
B = 8 \frac { 1 } { 2 } \
C = - 1
\end{array}$ 
D)  $\begin{array} { l } 
A = - 8 \frac { 1 } { 2 } \
B = 8 \frac { 1 } { 2 } \
C = 1
\end{array}$ 
E)  $\begin{array} { l } 
A = - 8 \frac { 1 } { 2 } \
B = 8 \frac { 1 } { 2 } \
C = 2
\end{array}$ 
A)  $\begin{array} { l } 
A = 8 \frac { 1 } { 2 } \
B = 1 \
C = 8 \frac { 1 } { 2 }
\end{array}$ 
B)  $\begin{array} { l } 
A = - 8 \frac { 1 } { 2 } \
B = 8 \frac { 1 } { 2 } \
C = 1
\end{array}$ 
C)  $\begin{array} { l } 
A = 8 \frac { 1 } { 2 } \
B = 8 \frac { 1 } { 2 } \
C = - 1
\end{array}$ 
D)  $\begin{array} { l } 
A = - 8 \frac { 1 } { 2 } \
B = 8 \frac { 1 } { 2 } \
C = 1
\end{array}$ 
E)  $\begin{array} { l } 
A = - 8 \frac { 1 } { 2 } \
B = 8 \frac { 1 } { 2 } \
C = 2
\end{array}$

Question 7

Use a graph to determine whether the equation has at least one real root. $x ^ { 2 } - 3 x + 6.26 = 0$

Accepted Answer

A)  three real roots 
B)  two real roots 
C)  no real roots 
D)  one real root 
E)  It is not possible to make this determination by graphing the equation. 
A)  three real roots 
B)  two real roots 
C)  no real roots 
D)  one real root 
E)  It is not possible to make this determination by graphing the equation.

Question 8

Determine which of the following real numbers is a root of the equation. $x ^ { 2 } - 2 x - 3 = 0$

Accepted Answer

A)  $x = 1 + \sqrt { 4 }$ 
B)  $x = 1 - \sqrt { 4 }$ 
C)  $x = 1 - \sqrt { 5 }$ 
D)  $x = 0$ 
E)  $x = 1 - \sqrt { 6 }$ 
A)  $x = 1 + \sqrt { 4 }$ 
B)  $x = 1 - \sqrt { 4 }$ 
C)  $x = 1 - \sqrt { 5 }$ 
D)  $x = 0$ 
E)  $x = 1 - \sqrt { 6 }$

Question 9

List the distinct roots of the following equation. For both repeated and single roots, specify their multiplicity. Enter $\left( r _ { 1 } , m _ { 1 } \right) , \left( r _ { 2 } , m _ { 2 } \right)$ , ... where $r _ { 1 } , r _ { 2 }$ etc. are the roots of the polynomial and $m _ { 1 }$ is the multiplicity of $r _ { 1 } , m _ { 2 }$ is the multiplicity of $r _ { 2 }$ etc. $( x - 3 ) ( x - 2 ) ^ { 6 } ( x - 9 ) = 0$

Accepted Answer

A)  (3, 1), (2, 4), (9, 1) 
B)  (3, 1), (2, 1), (9, 1) 
C)  (3, 1), (2, 5), (9, 1) 
D)  (3, 1), (2, 6), (9, 1) 
E)  (3, 4), (2, 4), (9, 4) 
A)  (3, 1), (2, 4), (9, 1) 
B)  (3, 1), (2, 1), (9, 1) 
C)  (3, 1), (2, 5), (9, 1) 
D)  (3, 1), (2, 6), (9, 1) 
E)  (3, 4), (2, 4), (9, 4)

Question 10

Find the rational roots and solve the equation. 3x 3  - 19x 2  + 21x - 5 = 0

Accepted Answer

A)  $\frac { 1 } { 3 } , 2,5$ 
B)  $\pm \frac { 1 } { 3 } , \pm 1$ 
C)  $\frac { 1 } { 3 } , 1,5$ 
D)  $\frac { 2 } { 3 } , 1,5$ 
E)  $- \frac { 1 } { 3 } , 2,5$ 
A)  $\frac { 1 } { 3 } , 2,5$ 
B)  $\pm \frac { 1 } { 3 } , \pm 1$ 
C)  $\frac { 1 } { 3 } , 1,5$ 
D)  $\frac { 2 } { 3 } , 1,5$ 
E)  $- \frac { 1 } { 3 } , 2,5$

Question 11

Factor the denominator of the given rational expression, and then find the partial fraction decomposition. $\frac { 2 x ^ { 3 } + 11 x - 3 } { x ^ { 4 } + 6 x ^ { 2 } + 9 }$

Accepted Answer

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

Question 12

Express $f ( x )$ in the form $a _ { n } x ^ { n } + a _ { n - 1 } x ^ { n - 1 } + \ldots + a _ { 1 } x + a _ { 0 }$ . Find a quadratic function that has a maximum value of 2 and that has - 2 and 4 as zeros.

Accepted Answer

A)  $f ( x ) = - \frac { 4 } { 9 } x ^ { 2 } + \frac { 2 } { 9 } x - \frac { 16 } { 9 }$ 
B)  $f ( x ) = - \frac { 9 } { 2 } x ^ { 2 } - \frac { 16 } { 9 } x + \frac { 4 } { 9 }$ 
C)  $f ( x ) = - \frac { 2 } { 9 } x ^ { 2 } + \frac { 4 } { 9 } x + \frac { 16 } { 9 }$ 
D)  $f ( x ) = \frac { 2 } { 9 } x ^ { 2 } - \frac { 9 } { 4 } x - \frac { 16 } { 9 }$ 
A)  $f ( x ) = - \frac { 4 } { 9 } x ^ { 2 } + \frac { 2 } { 9 } x - \frac { 16 } { 9 }$ 
B)  $f ( x ) = - \frac { 9 } { 2 } x ^ { 2 } - \frac { 16 } { 9 } x + \frac { 4 } { 9 }$ 
C)  $f ( x ) = - \frac { 2 } { 9 } x ^ { 2 } + \frac { 4 } { 9 } x + \frac { 16 } { 9 }$ 
D)  $f ( x ) = \frac { 2 } { 9 } x ^ { 2 } - \frac { 9 } { 4 } x - \frac { 16 } { 9 }$

Question 13

Use long division to find the quotient and the remainder. $\frac { x ^ { 3 } - x ^ { 2 } + 3 x - 6 } { x - 2 }$

Accepted Answer

A)  $\text { quotient: } x ^ { 2 } - x - 5 \text {; remainder: } 2$ 
B)  $\text { quotient: } x ^ { 2 } + x + 5 \text {; remainder: } 14$ 
C)  $\text { quotient: } x ^ { 2 } + x + 2 \text {; remainder: } - 5$ 
D)  $\text { quotient: } x ^ { 2 } + x + 5 \text {; remainder: } 4$ 
E)  $\text { quotient: } x ^ { 2 } + x - 5 \text {; remainder: } 6$ 
A)  $\text { quotient: } x ^ { 2 } - x - 5 \text {; remainder: } 2$ 
B)  $\text { quotient: } x ^ { 2 } + x + 5 \text {; remainder: } 14$ 
C)  $\text { quotient: } x ^ { 2 } + x + 2 \text {; remainder: } - 5$ 
D)  $\text { quotient: } x ^ { 2 } + x + 5 \text {; remainder: } 4$ 
E)  $\text { quotient: } x ^ { 2 } + x - 5 \text {; remainder: } 6$

Question 14

Use long division to find the quotient and the remainder. $\frac { 6 x ^ { 3 } - 3 x + 3 } { 2 x + 1 }$

Accepted Answer

A)  $\text { quotient: } 3 x ^ { 2 } - \frac { 3 } { 2 } x - \frac { 3 } { 4 } ; \text { remainder: } \frac { 15 } { 4 }$ 
B)  $\text { quotient: } 3 x ^ { 2 } - \frac { 3 } { 2 } x + \frac { 1 } { 4 } ; \text { remainder: } \frac { 15 } { 4 }$ 
C)  $\text { quotient: } 15 x ^ { 2 } + \frac { 3 } { 2 } x - \frac { 1 } { 4 } ; \text { remainder: } \frac { 3 } { 4 }$ 
D)  $\text { quotient: } 3 x ^ { 2 } - \frac { 3 } { 2 } x - \frac { 3 } { 4 } ; \text { remainder: } - \frac { 15 } { 4 }$ 
E)  $\text { quotient: } 3 x ^ { 2 } - \frac { 3 } { 2 } x - \frac { 3 } { 4 } ; \text { remainder: } \frac { 7 } { 4 }$ 
A)  $\text { quotient: } 3 x ^ { 2 } - \frac { 3 } { 2 } x - \frac { 3 } { 4 } ; \text { remainder: } \frac { 15 } { 4 }$ 
B)  $\text { quotient: } 3 x ^ { 2 } - \frac { 3 } { 2 } x + \frac { 1 } { 4 } ; \text { remainder: } \frac { 15 } { 4 }$ 
C)  $\text { quotient: } 15 x ^ { 2 } + \frac { 3 } { 2 } x - \frac { 1 } { 4 } ; \text { remainder: } \frac { 3 } { 4 }$ 
D)  $\text { quotient: } 3 x ^ { 2 } - \frac { 3 } { 2 } x - \frac { 3 } { 4 } ; \text { remainder: } - \frac { 15 } { 4 }$ 
E)  $\text { quotient: } 3 x ^ { 2 } - \frac { 3 } { 2 } x - \frac { 3 } { 4 } ; \text { remainder: } \frac { 7 } { 4 }$

Question 15

Determine the constants (denoted by capital letters) so that the equation is an identity. $\frac { 9 x - 14 } { ( x - 2 ) ( x + 2 ) } = \frac { A } { x - 2 } + \frac { B } { x + 2 }$

Accepted Answer

A)  $\begin{array} { l } 
A = 1 \
B = 7
\end{array}$ 
B)  $\begin{array} { l } 
A = - 1 \
B = 8
\end{array}$ 
C)  $\begin{array} { l } 
A = 2 \
B = 8
\end{array}$ 
D)  $\begin{array} { l } 
A = 8 \
B = 1
\end{array}$ 
E)  $\begin{array} { l } 
A = 1 \
B = 8
\end{array}$ 
A)  $\begin{array} { l } 
A = 1 \
B = 7
\end{array}$ 
B)  $\begin{array} { l } 
A = - 1 \
B = 8
\end{array}$ 
C)  $\begin{array} { l } 
A = 2 \
B = 8
\end{array}$ 
D)  $\begin{array} { l } 
A = 8 \
B = 1
\end{array}$ 
E)  $\begin{array} { l } 
A = 1 \
B = 8
\end{array}$

Question 16

Factor the denominator of the given rational expression, and then find the partial fraction decomposition. $\frac { 1 } { x ^ { 3 } + x ^ { 2 } - 44 x + 96 }$

Accepted Answer

A)  $\frac { \frac { 1 } { 11 } } { x - 3 } + \frac { - \frac { 1 } { 12 } } { x - 4 } + \frac { \frac { 1 } { 132 } } { x + 8 }$ 
B)  $\frac { - \frac { 1 } { 11 } } { x - 3 } + \frac { - \frac { 1 } { 12 } } { x - 4 } + \frac { \frac { 1 } { 132 } } { x + 8 }$ 
C)  $\frac { \frac { 1 } { 11 } } { x - 3 } + \frac { - \frac { 1 } { 12 } } { x - 4 } + \frac { - \frac { 1 } { 132 } } { x + 8 }$ 
D)  $\frac { - \frac { 1 } { 11 } } { x - 3 } + \frac { \frac { 1 } { 12 } } { x - 4 } + \frac { \frac { 1 } { 132 } } { x + 8 }$ 
E)  $\frac { - \frac { 1 } { 11 } } { x - 3 } + \frac { \frac { 1 } { 12 } } { x - 4 } + \frac { - \frac { 1 } { 132 } } { x + 8 }$ 
A)  $\frac { \frac { 1 } { 11 } } { x - 3 } + \frac { - \frac { 1 } { 12 } } { x - 4 } + \frac { \frac { 1 } { 132 } } { x + 8 }$ 
B)  $\frac { - \frac { 1 } { 11 } } { x - 3 } + \frac { - \frac { 1 } { 12 } } { x - 4 } + \frac { \frac { 1 } { 132 } } { x + 8 }$ 
C)  $\frac { \frac { 1 } { 11 } } { x - 3 } + \frac { - \frac { 1 } { 12 } } { x - 4 } + \frac { - \frac { 1 } { 132 } } { x + 8 }$ 
D)  $\frac { - \frac { 1 } { 11 } } { x - 3 } + \frac { \frac { 1 } { 12 } } { x - 4 } + \frac { \frac { 1 } { 132 } } { x + 8 }$ 
E)  $\frac { - \frac { 1 } { 11 } } { x - 3 } + \frac { \frac { 1 } { 12 } } { x - 4 } + \frac { - \frac { 1 } { 132 } } { x + 8 }$

Question 17

Use long division to find the quotient and the remainder. $\frac { 2 x ^ { 6 } + 2 } { x - 1 }$

Accepted Answer

A)  $\text { quotient: } 2 x ^ { 5 } - 2 x ^ { 4 } + 2 x ^ { 3 } - 2 x ^ { 2 } + 2 x - 2 \text {; remainder: } 0$ 
B)  $\text { quotient: } 2 x ^ { 5 } + 2 x ^ { 4 } + 2 x ^ { 3 } + 2 x ^ { 2 } + 2 x + 2 \text {; remainder: } 4$ 
C)  $\text { quotient: } 2 x ^ { 5 } + 2 x ^ { 3 } + 2 x + 2 ; \text { remainder: } 2$ 
D)  $\text { quotient: } 2 x ^ { 5 } + 2 x ^ { 4 } - 2 x ^ { 3 } + 2 x ^ { 2 } - 2 x + 2 \text {; remainder: } 4$ 
E)  $\text { quotient: } 2 x ^ { 5 } + 2 x ^ { 4 } + 2 x ^ { 3 } + 2 x ^ { 2 } + 2 x + 2 \text {; remainder: } 0$ 
A)  $\text { quotient: } 2 x ^ { 5 } - 2 x ^ { 4 } + 2 x ^ { 3 } - 2 x ^ { 2 } + 2 x - 2 \text {; remainder: } 0$ 
B)  $\text { quotient: } 2 x ^ { 5 } + 2 x ^ { 4 } + 2 x ^ { 3 } + 2 x ^ { 2 } + 2 x + 2 \text {; remainder: } 4$ 
C)  $\text { quotient: } 2 x ^ { 5 } + 2 x ^ { 3 } + 2 x + 2 ; \text { remainder: } 2$ 
D)  $\text { quotient: } 2 x ^ { 5 } + 2 x ^ { 4 } - 2 x ^ { 3 } + 2 x ^ { 2 } - 2 x + 2 \text {; remainder: } 4$ 
E)  $\text { quotient: } 2 x ^ { 5 } + 2 x ^ { 4 } + 2 x ^ { 3 } + 2 x ^ { 2 } + 2 x + 2 \text {; remainder: } 0$

Question 18

Find the rational roots and solve the equation. 4x 3  + x 2  - 20x - 5 = 0

Accepted Answer

A)  $\pm 5 , - \frac { 1 } { 4 }$ 
B)  $- \frac { 1 } { \sqrt { 5 } } , \pm 4$ 
C)  $\pm \sqrt { 5 } , - 4$ 
D)  $\pm \sqrt { 5 } , - \frac { 1 } { 4 }$ 
E)  $\pm \sqrt { 5 } , \frac { 1 } { 4 }$ 
A)  $\pm 5 , - \frac { 1 } { 4 }$ 
B)  $- \frac { 1 } { \sqrt { 5 } } , \pm 4$ 
C)  $\pm \sqrt { 5 } , - 4$ 
D)  $\pm \sqrt { 5 } , - \frac { 1 } { 4 }$ 
E)  $\pm \sqrt { 5 } , \frac { 1 } { 4 }$

Question 19

Use synthetic division to find the quotient and the remainder. $\frac { 4 x ^ { 4 } + 16 x ^ { 3 } + 24 x ^ { 2 } + 16 x + 4 } { x + 1 }$

Accepted Answer

A)  $\text { quotient: } 4 x ^ { 3 } + 20 x ^ { 2 } + 12 x + 4 \text {; remainder: } 0$ 
B)  $\text { quotient: } 4 x ^ { 3 } + 12 x ^ { 2 } + 12 x + 4 \text {; remainder: } 0$ 
C)  $\text { quotient: } 4 x ^ { 3 } + 4 x ^ { 2 } + 12 x - 12 \text {; remainder: } 3$ 
D)  $\text { quotient: } 4 x ^ { 3 } + 12 x ^ { 2 } + 12 x + 4 \text {; remainder: } 3$ 
E)  $\text { quotient: } 4 x ^ { 3 } - 12 x ^ { 2 } + 4 x + 12 \text {; remainder: } 0$ 
A)  $\text { quotient: } 4 x ^ { 3 } + 20 x ^ { 2 } + 12 x + 4 \text {; remainder: } 0$ 
B)  $\text { quotient: } 4 x ^ { 3 } + 12 x ^ { 2 } + 12 x + 4 \text {; remainder: } 0$ 
C)  $\text { quotient: } 4 x ^ { 3 } + 4 x ^ { 2 } + 12 x - 12 \text {; remainder: } 3$ 
D)  $\text { quotient: } 4 x ^ { 3 } + 12 x ^ { 2 } + 12 x + 4 \text {; remainder: } 3$ 
E)  $\text { quotient: } 4 x ^ { 3 } - 12 x ^ { 2 } + 4 x + 12 \text {; remainder: } 0$

Question 20

Factor the denominator of the given rational expression, and then find the partial fraction decomposition. $\frac { 11 x - 3 \sqrt { 3 } } { x ^ { 2 } - 3 }$

Accepted Answer

A)  $\frac { 11 x } { x + \sqrt { 3 } } + \frac { - 3 \sqrt { 3 } } { x - \sqrt { 3 } }$ 
B)  $\frac { 4 } { x + \sqrt { 3 } } + \frac { 7 } { x - \sqrt { 3 } }$ 
C)  $\frac { 7 } { x + 3 } + \frac { 4 } { x - 3 }$ 
D)  $\frac { 7 } { x + \sqrt { 3 } } + \frac { - 4 } { x - \sqrt { 3 } }$ 
E)  $\frac { 7 } { x + \sqrt { 3 } } + \frac { 4 } { x - \sqrt { 3 } }$ 
A)  $\frac { 11 x } { x + \sqrt { 3 } } + \frac { - 3 \sqrt { 3 } } { x - \sqrt { 3 } }$ 
B)  $\frac { 4 } { x + \sqrt { 3 } } + \frac { 7 } { x - \sqrt { 3 } }$ 
C)  $\frac { 7 } { x + 3 } + \frac { 4 } { x - 3 }$ 
D)  $\frac { 7 } { x + \sqrt { 3 } } + \frac { - 4 } { x - \sqrt { 3 } }$ 
E)  $\frac { 7 } { x + \sqrt { 3 } } + \frac { 4 } { x - \sqrt { 3 } }$

Determine upper and lower bounds for the real roots of the equation. 5x⁴ - 10x - 12 = 0

Determine the constants (denoted with capital letters) so that the equation is an identity. $\frac { 8 x + 3 } { ( x + 3 ) ^ { 2 } } = \frac { A } { x + 3 } + \frac { B } { ( x + 3 ) ^ { 2 } }$

Determine which of the following polynomials is not reducible.

Find a quadratic equation with rational coefficients, one of whose roots is the given number. Write your answer so that the coefficient of $x ^ { 2 }$ is 1. $r _ { 1 } = 1 + \sqrt { 2 }$

Use Descartes's rule of signs to obtain information regarding the roots of the equation. $x ^ { 9 } + 5 = 0$

Determine the constants (denoted by capital letters) so that the equation is an identity. $\frac { x - 17 } { x \left( x ^ { 2 } + 2 \right) } = \frac { A } { x } + \frac { B x + C } { x ^ { 2 } + 2 }$

Use a graph to determine whether the equation has at least one real root. $x ^ { 2 } - 3 x + 6.26 = 0$

Determine which of the following real numbers is a root of the equation. $x ^ { 2 } - 2 x - 3 = 0$

Find the rational roots and solve the equation. 3x³ - 19x² + 21x - 5 = 0

Factor the denominator of the given rational expression, and then find the partial fraction decomposition. $\frac { 2 x ^ { 3 } + 11 x - 3 } { x ^ { 4 } + 6 x ^ { 2 } + 9 }$

Express $f ( x )$ in the form $a _ { n } x ^ { n } + a _ { n - 1 } x ^ { n - 1 } + \ldots + a _ { 1 } x + a _ { 0 }$ . Find a quadratic function that has a maximum value of 2 and that has - 2 and 4 as zeros.

Use long division to find the quotient and the remainder. $\frac { x ^ { 3 } - x ^ { 2 } + 3 x - 6 } { x - 2 }$

Use long division to find the quotient and the remainder. $\frac { 6 x ^ { 3 } - 3 x + 3 } { 2 x + 1 }$

Determine the constants (denoted by capital letters) so that the equation is an identity. $\frac { 9 x - 14 } { ( x - 2 ) ( x + 2 ) } = \frac { A } { x - 2 } + \frac { B } { x + 2 }$

Factor the denominator of the given rational expression, and then find the partial fraction decomposition. $\frac { 1 } { x ^ { 3 } + x ^ { 2 } - 44 x + 96 }$

Use long division to find the quotient and the remainder. $\frac { 2 x ^ { 6 } + 2 } { x - 1 }$

Find the rational roots and solve the equation. 4x³ + x² - 20x - 5 = 0

Use synthetic division to find the quotient and the remainder. $\frac { 4 x ^ { 4 } + 16 x ^ { 3 } + 24 x ^ { 2 } + 16 x + 4 } { x + 1 }$

Factor the denominator of the given rational expression, and then find the partial fraction decomposition. $\frac { 11 x - 3 \sqrt { 3 } } { x ^ { 2 } - 3 }$

Fundamentals

Equations and Inequalities

Functions

Polynomial and Rational Functions Applications to Optimization

Exponential and Logarithmic Functions

An Introduction to Trigonometry Via Right Triangles

The Trigonometric Functions

Graphs of the Trigonometric Functions

Analytical Trigonometry

Additional Topics in Trigonometry

Systems of Equations

The Conic Sections

Additional Topics in Algebra

Filters

Exam 13: Roots of Polynomial Equations

Determine upper and lower bounds for the real roots of the equation. 5x 4 - 10x - 12 = 0

Determine the constants (denoted with capital letters) so that the equation is an identity. 8x+3(x+3)2=Ax+3+B(x+3)2\frac { 8 x + 3 } { ( x + 3 ) ^ { 2 } } = \frac { A } { x + 3 } + \frac { B } { ( x + 3 ) ^ { 2 } }(x+3)28x+3​=x+3A​+(x+3)2B​

Determine which of the following polynomials is not reducible.

Find a quadratic equation with rational coefficients, one of whose roots is the given number. Write your answer so that the coefficient of x2x ^ { 2 }x2 is 1. r1=1+2r _ { 1 } = 1 + \sqrt { 2 }r1​=1+2​

Use Descartes's rule of signs to obtain information regarding the roots of the equation. x9+5=0x ^ { 9 } + 5 = 0x9+5=0

Determine the constants (denoted by capital letters) so that the equation is an identity. x−17x(x2+2)=Ax+Bx+Cx2+2\frac { x - 17 } { x \left( x ^ { 2 } + 2 \right) } = \frac { A } { x } + \frac { B x + C } { x ^ { 2 } + 2 }x(x2+2)x−17​=xA​+x2+2Bx+C​

Use a graph to determine whether the equation has at least one real root. x2−3x+6.26=0x ^ { 2 } - 3 x + 6.26 = 0x2−3x+6.26=0

Determine which of the following real numbers is a root of the equation. x2−2x−3=0x ^ { 2 } - 2 x - 3 = 0x2−2x−3=0

Find the rational roots and solve the equation. 3x 3 - 19x 2 + 21x - 5 = 0

Factor the denominator of the given rational expression, and then find the partial fraction decomposition. 2x3+11x−3x4+6x2+9\frac { 2 x ^ { 3 } + 11 x - 3 } { x ^ { 4 } + 6 x ^ { 2 } + 9 }x4+6x2+92x3+11x−3​

Express f(x)f ( x )f(x) in the form anxn+an−1xn−1+…+a1x+a0a _ { n } x ^ { n } + a _ { n - 1 } x ^ { n - 1 } + \ldots + a _ { 1 } x + a _ { 0 }an​xn+an−1​xn−1+…+a1​x+a0​ . Find a quadratic function that has a maximum value of 2 and that has - 2 and 4 as zeros.

Use long division to find the quotient and the remainder. x3−x2+3x−6x−2\frac { x ^ { 3 } - x ^ { 2 } + 3 x - 6 } { x - 2 }x−2x3−x2+3x−6​

Use long division to find the quotient and the remainder. 6x3−3x+32x+1\frac { 6 x ^ { 3 } - 3 x + 3 } { 2 x + 1 }2x+16x3−3x+3​

Determine the constants (denoted by capital letters) so that the equation is an identity. 9x−14(x−2)(x+2)=Ax−2+Bx+2\frac { 9 x - 14 } { ( x - 2 ) ( x + 2 ) } = \frac { A } { x - 2 } + \frac { B } { x + 2 }(x−2)(x+2)9x−14​=x−2A​+x+2B​

Factor the denominator of the given rational expression, and then find the partial fraction decomposition. 1x3+x2−44x+96\frac { 1 } { x ^ { 3 } + x ^ { 2 } - 44 x + 96 }x3+x2−44x+961​

Use long division to find the quotient and the remainder. 2x6+2x−1\frac { 2 x ^ { 6 } + 2 } { x - 1 }x−12x6+2​

Find the rational roots and solve the equation. 4x 3 + x 2 - 20x - 5 = 0

Use synthetic division to find the quotient and the remainder. 4x4+16x3+24x2+16x+4x+1\frac { 4 x ^ { 4 } + 16 x ^ { 3 } + 24 x ^ { 2 } + 16 x + 4 } { x + 1 }x+14x4+16x3+24x2+16x+4​

Factor the denominator of the given rational expression, and then find the partial fraction decomposition. 11x−33x2−3\frac { 11 x - 3 \sqrt { 3 } } { x ^ { 2 } - 3 }x2−311x−33​​

Fundamentals

Equations and Inequalities

Functions

Polynomial and Rational Functions Applications to Optimization

Exponential and Logarithmic Functions

An Introduction to Trigonometry Via Right Triangles

The Trigonometric Functions

Graphs of the Trigonometric Functions

Analytical Trigonometry

Additional Topics in Trigonometry

Systems of Equations

The Conic Sections

Additional Topics in Algebra

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Determine upper and lower bounds for the real roots of the equation. 5x⁴ - 10x - 12 = 0

Determine the constants (denoted with capital letters) so that the equation is an identity. $\frac { 8 x + 3 } { ( x + 3 ) ^ { 2 } } = \frac { A } { x + 3 } + \frac { B } { ( x + 3 ) ^ { 2 } }$

Find a quadratic equation with rational coefficients, one of whose roots is the given number. Write your answer so that the coefficient of $x ^ { 2 }$ is 1. $r _ { 1 } = 1 + \sqrt { 2 }$

Use Descartes's rule of signs to obtain information regarding the roots of the equation. $x ^ { 9 } + 5 = 0$

Determine the constants (denoted by capital letters) so that the equation is an identity. $\frac { x - 17 } { x \left( x ^ { 2 } + 2 \right) } = \frac { A } { x } + \frac { B x + C } { x ^ { 2 } + 2 }$

Use a graph to determine whether the equation has at least one real root. $x ^ { 2 } - 3 x + 6.26 = 0$

Determine which of the following real numbers is a root of the equation. $x ^ { 2 } - 2 x - 3 = 0$

Find the rational roots and solve the equation. 3x³ - 19x² + 21x - 5 = 0

Factor the denominator of the given rational expression, and then find the partial fraction decomposition. $\frac { 2 x ^ { 3 } + 11 x - 3 } { x ^ { 4 } + 6 x ^ { 2 } + 9 }$

Express $f ( x )$ in the form $a _ { n } x ^ { n } + a _ { n - 1 } x ^ { n - 1 } + \ldots + a _ { 1 } x + a _ { 0 }$ . Find a quadratic function that has a maximum value of 2 and that has - 2 and 4 as zeros.

Use long division to find the quotient and the remainder. $\frac { x ^ { 3 } - x ^ { 2 } + 3 x - 6 } { x - 2 }$

Use long division to find the quotient and the remainder. $\frac { 6 x ^ { 3 } - 3 x + 3 } { 2 x + 1 }$

Determine the constants (denoted by capital letters) so that the equation is an identity. $\frac { 9 x - 14 } { ( x - 2 ) ( x + 2 ) } = \frac { A } { x - 2 } + \frac { B } { x + 2 }$

Factor the denominator of the given rational expression, and then find the partial fraction decomposition. $\frac { 1 } { x ^ { 3 } + x ^ { 2 } - 44 x + 96 }$

Use long division to find the quotient and the remainder. $\frac { 2 x ^ { 6 } + 2 } { x - 1 }$

Find the rational roots and solve the equation. 4x³ + x² - 20x - 5 = 0

Use synthetic division to find the quotient and the remainder. $\frac { 4 x ^ { 4 } + 16 x ^ { 3 } + 24 x ^ { 2 } + 16 x + 4 } { x + 1 }$

Factor the denominator of the given rational expression, and then find the partial fraction decomposition. $\frac { 11 x - 3 \sqrt { 3 } } { x ^ { 2 } - 3 }$