Question 1

Find all the real zeros of the polynomial function. $f ( x ) = x ^ { 5 } - 10 x ^ { 3 } + 25 x$

Accepted Answer

A)  $0 , \pm \sqrt { 5 }$ 
B)  0,5 
C)  $0 , - \sqrt { 5 }$ 
D)  $0 , - 5$ 
E)  $0 , \sqrt { 5 }$ 
A)  $0 , \pm \sqrt { 5 }$ 
B)  0,5 
C)  $0 , - \sqrt { 5 }$ 
D)  $0 , - 5$ 
E)  $0 , \sqrt { 5 }$

Question 2

Select the correct graph of the following function. $$f ( x ) = \frac { x ^ { 2 } } { 6 x + 1 }$$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
E) 
  
A) 
  
B) 
  
C) 
  
D) 
  
E)

Question 3

Use the Remainder Theorem and synthetic division to find the function value. Verify your answers using another method. 
$h ( x ) = x ^ { 3 } - 6 x ^ { 2 } - 7 x + 5 , \quad h ( - 6 )$

Accepted Answer

A)  -387 
B)  -385 
C)  -381 
D)  -384 
E)  -383 
A)  -387 
B)  -385 
C)  -381 
D)  -384 
E)  -383

Question 4

Select the graph of.$f$ 
$f ( x ) = x ^ { 3 } + x ^ { 2 } - 4 x - 4$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
E) 
  
A) 
  
B) 
  
C) 
  
D) 
  
E)

Question 5

Write the quadratic function $f ( x ) = - x ^ { 2 } + 2 x + 2$ in standard form.

Accepted Answer

A)  $f ( x ) = ( x - 3 ) ^ { 2 } - 1$ 
B)  $f ( x ) = ( x - 1 ) ^ { 2 } - 3$ 
C)  $f ( x ) = - ( x + 1 ) ^ { 2 } + 3$ 
D)  $f ( x ) = - ( x + 3 ) ^ { 2 } - 1$ 
E)  $f ( x ) = - ( x - 1 ) ^ { 2 } + 3$ 
A)  $f ( x ) = ( x - 3 ) ^ { 2 } - 1$ 
B)  $f ( x ) = ( x - 1 ) ^ { 2 } - 3$ 
C)  $f ( x ) = - ( x + 1 ) ^ { 2 } + 3$ 
D)  $f ( x ) = - ( x + 3 ) ^ { 2 } - 1$ 
E)  $f ( x ) = - ( x - 1 ) ^ { 2 } + 3$

Question 6

Use Descartes's Rule of Signs to determine the possible numbers of positive and neg- ative zeros of the function. $5 x ^ { 5 } - 10 x$

Accepted Answer

A)  One positive zero, one negative zero 
B)  Two negative zeros or no positive zero 
C)  One negative zero 
D)  One negative zero, four negative zeros 
E)  One negative zero or three positive zeros 
A)  One positive zero, one negative zero 
B)  Two negative zeros or no positive zero 
C)  One negative zero 
D)  One negative zero, four negative zeros 
E)  One negative zero or three positive zeros

Question 7

Find the domain of $x$ in the expression. Use a graphing utility to verify your result. 
$\sqrt { 25 - 16 x ^ { 2 } }$

Accepted Answer

A)  $\left[ - \frac { 5 } { 4 } , - \frac { 5 } { 4 } \right]$ 
B)  $\left[ - \frac { 4 } { 5 } , - \frac { 4 } { 5 } \right]$ 
C)  $\left[ - \frac { 4 } { 5 } , \frac { 4 } { 5 } \right]$ 
D)  $\left[ \frac { 5 } { 4 } , \frac { 5 } { 4 } \right]$ 
E)  $\left[ - \frac { 5 } { 4 } , \frac { 5 } { 4 } \right]$ 
A)  $\left[ - \frac { 5 } { 4 } , - \frac { 5 } { 4 } \right]$ 
B)  $\left[ - \frac { 4 } { 5 } , - \frac { 4 } { 5 } \right]$ 
C)  $\left[ - \frac { 4 } { 5 } , \frac { 4 } { 5 } \right]$ 
D)  $\left[ \frac { 5 } { 4 } , \frac { 5 } { 4 } \right]$ 
E)  $\left[ - \frac { 5 } { 4 } , \frac { 5 } { 4 } \right]$

Question 8

Perform the addition or subtraction and write the result in standard form. $( 9 + i ) + ( 2 - 6 i )$

Accepted Answer

A)  $12 - 6 i$ 
B)  $14 - 8 i$ 
C)  $13 - 7 i$ 
D)  $11 - 5 i$ 
E)  $15 - 9 i$ 
A)  $12 - 6 i$ 
B)  $14 - 8 i$ 
C)  $13 - 7 i$ 
D)  $11 - 5 i$ 
E)  $15 - 9 i$

Question 9

Perform the addition or subtraction and write the result in standard form. $$17 i - ( 18 - 8 i )$$

Accepted Answer

A)  $- 18 + 25 i$ 
B)  $18 + 25 i$ 
C)  $- 18 + 28 i$ 
D)  $- 18 - 25 i$ 
E)  $18 - 25 i$ 
A)  $- 18 + 25 i$ 
B)  $18 + 25 i$ 
C)  $- 18 + 28 i$ 
D)  $- 18 - 25 i$ 
E)  $18 - 25 i$

Question 10

Perform the operation and write the result in standard form. $\frac { 5 } { 1 + i } - \frac { 6 } { 1 - i }$

Accepted Answer

A)  $- \frac { 1 } { 2 } - \frac { 2 } { 11 } i$ 
B)  $- \frac { 1 } { 2 } - \frac { 11 } { 2 } i$ 
C)  $- \frac { 1 } { 2 } + \frac { 2 } { 11 } i$ 
D)  $- \frac { 1 } { 2 } + \frac { 11 } { 2 } i$ 
E)  $\frac { 11 } { 2 } - \frac { 1 } { 2 } i$ 
A)  $- \frac { 1 } { 2 } - \frac { 2 } { 11 } i$ 
B)  $- \frac { 1 } { 2 } - \frac { 11 } { 2 } i$ 
C)  $- \frac { 1 } { 2 } + \frac { 2 } { 11 } i$ 
D)  $- \frac { 1 } { 2 } + \frac { 11 } { 2 } i$ 
E)  $\frac { 11 } { 2 } - \frac { 1 } { 2 } i$

Question 11

Simplify the complex number and write it in standard form. $( - i ) ^ { 5 }$

Accepted Answer

A)  5 
B)  $i$ 
C)  1 
D)  $- 1$ 
E)  $- i$ 
A)  5 
B)  $i$ 
C)  1 
D)  $- 1$ 
E)  $- i$

Question 12

Write the complex number in standard form. $$( 3 + \sqrt { - 6 } ) ( 6 - \sqrt { - 6 } )$$

Accepted Answer

A)  $( 18 + 6 ) - ( 6 \sqrt { 6 } - 3 \sqrt { 6 } ) i$ 
B)  $( 18 + 6 ) + ( 6 \sqrt { 6 } + 3 \sqrt { 6 } ) i$ 
C)  $( 18 - 6 ) - ( 6 \sqrt { 6 } - 3 \sqrt { 6 } ) i$ 
D)  $( 18 - 6 ) + ( 6 \sqrt { 6 } - 3 \sqrt { 6 } ) i$ 
E)  $( 18 + 6 ) + ( 6 \sqrt { 6 } - 3 \sqrt { 6 } ) i$ 
A)  $( 18 + 6 ) - ( 6 \sqrt { 6 } - 3 \sqrt { 6 } ) i$ 
B)  $( 18 + 6 ) + ( 6 \sqrt { 6 } + 3 \sqrt { 6 } ) i$ 
C)  $( 18 - 6 ) - ( 6 \sqrt { 6 } - 3 \sqrt { 6 } ) i$ 
D)  $( 18 - 6 ) + ( 6 \sqrt { 6 } - 3 \sqrt { 6 } ) i$ 
E)  $( 18 + 6 ) + ( 6 \sqrt { 6 } - 3 \sqrt { 6 } ) i$

Question 13

Use long division to divide. 
$\left( x ^ { 4 } + 4 x ^ { 2 } + 5 \right) \div \left( x ^ { 2 } + x + 4 \right)$

Accepted Answer

A)  $x ^ { 2 } - x + 1 - \frac { 4 } { x ^ { 2 } - x + 1 }$ 
B)  $x ^ { 2 } - x + 1 + \frac { 3 x + 1 } { x ^ { 2 } + x + 4 }$ 
C)  $x ^ { 2 } - x + 1$ 
D)  $x ^ { 2 } + x - 1$ 
E)  $x ^ { 2 } + x - 1 + \frac { 3 x - 4 } { x ^ { 2 } + x + 4 }$ 
A)  $x ^ { 2 } - x + 1 - \frac { 4 } { x ^ { 2 } - x + 1 }$ 
B)  $x ^ { 2 } - x + 1 + \frac { 3 x + 1 } { x ^ { 2 } + x + 4 }$ 
C)  $x ^ { 2 } - x + 1$ 
D)  $x ^ { 2 } + x - 1$ 
E)  $x ^ { 2 } + x - 1 + \frac { 3 x - 4 } { x ^ { 2 } + x + 4 }$

Question 14

Simplify the rational expression, $\frac { x ^ { 4 } + x ^ { 3 } - 21 x ^ { 2 } - 41 x - 20 } { x ^ { 2 } + 2 x + 1 }$ ,by using long division or synthetic division.

Accepted Answer

A)  $x ^ { 2 } - x - 20$ 
B)  $x ^ { 2 } - 9 x + 25$ 
C)  $- x ^ { 2 } + 8 x + 25$ 
D)  $x ^ { 2 } + 5 x + 4$ 
E)  $- x ^ { 2 } + 9 x - 20$ 
A)  $x ^ { 2 } - x - 20$ 
B)  $x ^ { 2 } - 9 x + 25$ 
C)  $- x ^ { 2 } + 8 x + 25$ 
D)  $x ^ { 2 } + 5 x + 4$ 
E)  $- x ^ { 2 } + 9 x - 20$

Question 15

Select the graph of the function $y = - 3 x ^ { 2 }$ . Compare the graph of this function with the graph of $y = x ^ { 2 }$

Accepted Answer

A) 
 
Vertical stretch 
B)

Vertical stretch and reflection in $x$-axis 
C) 
 
 Reflection in $X$ -axis 
D) 
 
Vertical shrink and reflection in $x$-axis 
E) 
 
Vertical shrink 
A) 
 
Vertical stretch 
B)

Vertical stretch and reflection in $x$-axis 
C) 
 
 Reflection in $X$ -axis 
D) 
 
Vertical shrink and reflection in $x$-axis 
E) 
 
Vertical shrink

Question 16

Find all the zeros of the function and write the polynomial as a product of linear fac- tors. 
$y ^ { 4 } - 2401$

Accepted Answer

A)  $\pm 7 ( y - 7 ) ( y + 7 )$ 
B)  $\pm 7 , \pm 7 i ; ( y - 7 ) ( y + 7 ) ( y - 7 i ) ( y + 7 i )$ 
C)  $7,7 i ; ( y + 7 ) ( y + 7 ) ( y - 7 i ) ( y - 7 i )$ 
D)  $\pm 7 i ; ( y - 7 i ) ( y + 7 i )$ 
E)  $\pm 7 , \pm 7 i ; ( y - 7 ) ( y - 7 ) ( y + 7 i ) ( y + 7 i )$ 
A)  $\pm 7 ( y - 7 ) ( y + 7 )$ 
B)  $\pm 7 , \pm 7 i ; ( y - 7 ) ( y + 7 ) ( y - 7 i ) ( y + 7 i )$ 
C)  $7,7 i ; ( y + 7 ) ( y + 7 ) ( y - 7 i ) ( y - 7 i )$ 
D)  $\pm 7 i ; ( y - 7 i ) ( y + 7 i )$ 
E)  $\pm 7 , \pm 7 i ; ( y - 7 ) ( y - 7 ) ( y + 7 i ) ( y + 7 i )$

Question 17

Use the position equation $s = - 20 t ^ { 2 } + v _ { 0 } t + s _ { 0 }$ , where $S$ represents the height of an object (in feet), $\mathcal { V } _ { 0 }$ represents the initial velocity of the object (in feet per second), $s _ { 0 }$ represents the Initial height of the object (in feet), and $t$ represents the time (in seconds).
A projectile is fired straight upward from ground level $\left( s _ { 0 } = 0 \right)$ with an initial velocity of $220$ feet Per second. At what instant will it be back at ground level?

Accepted Answer

A)  $t = 13 \mathrm { Sec }$ 
B)  $t = 12 \mathrm { Sec }$ 
C)  $t = 20 \mathrm { Sec }$ 
D)  $t = 11 \mathrm { Sec }$ 
E)  $t = 10 \mathrm { Sec }$ 
A)  $t = 13 \mathrm { Sec }$ 
B)  $t = 12 \mathrm { Sec }$ 
C)  $t = 20 \mathrm { Sec }$ 
D)  $t = 11 \mathrm { Sec }$ 
E)  $t = 10 \mathrm { Sec }$

Question 18

Determine the equations of any horizontal and vertical asymptotes of $$f ( x ) = \frac { x ^ { 2 } + 8 x + 15 } { x ^ { 3 } + 5 x ^ { 2 } - x - 5 }$$

Accepted Answer

A)  horizontal: $y = 1$; vertical: $x = - 5$ 
B)  horizontal: $y = 0$; vertical: $x = 1$ and $x = - 5$ 
C)  horizontal: $y = 0$; vertical: $x = - 1$ and $x = 1$ 
D)  horizontal: none; vertical: none 
E)  horizontal: $y = 0$; vertical: $x = - 5$ and $x = - 3$ 
A)  horizontal: $y = 1$; vertical: $x = - 5$ 
B)  horizontal: $y = 0$; vertical: $x = 1$ and $x = - 5$ 
C)  horizontal: $y = 0$; vertical: $x = - 1$ and $x = 1$ 
D)  horizontal: none; vertical: none 
E)  horizontal: $y = 0$; vertical: $x = - 5$ and $x = - 3$

Question 19

Solve the equation and write complex solutions in standard form. $$x ^ { 2 } - 6 x + 58 = 0$$

Accepted Answer

A)  $x = 3 + 7 i , 3 - 7 i$ 
B)  $x = 7 + 9 i , 7 - 9 i$ 
C)  $x = 3 + 49 i , 3 - 49 i$ 
D)  $x = 21 + 7 i , 21 - 7 i$ 
E)  $x = 7 + 3 i , 7 - 3 i$ 
A)  $x = 3 + 7 i , 3 - 7 i$ 
B)  $x = 7 + 9 i , 7 - 9 i$ 
C)  $x = 3 + 49 i , 3 - 49 i$ 
D)  $x = 21 + 7 i , 21 - 7 i$ 
E)  $x = 7 + 3 i , 7 - 3 i$

Question 20

Select the graph of the function and determine the zeros of the polynomial. $$f ( x ) = x ^ { 3 } - 4 x ^ { 2 }$$

Accepted Answer

A)  0,4
  
B) 0,4
  
C) 4,-4
  
D) $0 , - 4$
  
E) 0,-4
  
A)  0,4
  
B) 0,4
  
C) 4,-4
  
D) $0 , - 4$
  
E) 0,-4

Find all the real zeros of the polynomial function. $f ( x ) = x ^ { 5 } - 10 x ^ { 3 } + 25 x$

Select the correct graph of the following function. $f ( x ) = \frac { x ^ { 2 } } { 6 x + 1 }$

Use the Remainder Theorem and synthetic division to find the function value. Verify your answers using another method. $h ( x ) = x ^ { 3 } - 6 x ^ { 2 } - 7 x + 5 , \quad h ( - 6 )$

Select the graph of. $f$ $f ( x ) = x ^ { 3 } + x ^ { 2 } - 4 x - 4$

Write the quadratic function $f ( x ) = - x ^ { 2 } + 2 x + 2$ in standard form.

Use Descartes's Rule of Signs to determine the possible numbers of positive and neg- ative zeros of the function. $5 x ^ { 5 } - 10 x$

Find the domain of $x$ in the expression. Use a graphing utility to verify your result. $\sqrt { 25 - 16 x ^ { 2 } }$

Perform the addition or subtraction and write the result in standard form. $( 9 + i ) + ( 2 - 6 i )$

Perform the addition or subtraction and write the result in standard form. $17 i - ( 18 - 8 i )$

Perform the operation and write the result in standard form. $\frac { 5 } { 1 + i } - \frac { 6 } { 1 - i }$

Simplify the complex number and write it in standard form. $( - i ) ^ { 5 }$

Write the complex number in standard form. $( 3 + \sqrt { - 6 } ) ( 6 - \sqrt { - 6 } )$

Use long division to divide. $\left( x ^ { 4 } + 4 x ^ { 2 } + 5 \right) \div \left( x ^ { 2 } + x + 4 \right)$

Simplify the rational expression, $\frac { x ^ { 4 } + x ^ { 3 } - 21 x ^ { 2 } - 41 x - 20 } { x ^ { 2 } + 2 x + 1 }$ ,by using long division or synthetic division.

Select the graph of the function $y = - 3 x ^ { 2 }$ . Compare the graph of this function with the graph of $y = x ^ { 2 }$

Find all the zeros of the function and write the polynomial as a product of linear fac- tors. $y ^ { 4 } - 2401$

Determine the equations of any horizontal and vertical asymptotes of $f ( x ) = \frac { x ^ { 2 } + 8 x + 15 } { x ^ { 3 } + 5 x ^ { 2 } - x - 5 }$

Solve the equation and write complex solutions in standard form. $x ^ { 2 } - 6 x + 58 = 0$

Select the graph of the function and determine the zeros of the polynomial. $f ( x ) = x ^ { 3 } - 4 x ^ { 2 }$

Functions and Their Graphs

Exponential and Logarithmic Functions

Trigonometry

Analytic Trigonometry

Topics in Analytic Geometry

Filters

Exam 2: Polynomial and Rational Functions

Find all the real zeros of the polynomial function. f(x)=x5−10x3+25xf ( x ) = x ^ { 5 } - 10 x ^ { 3 } + 25 xf(x)=x5−10x3+25x

Select the correct graph of the following function. f(x)=x26x+1f ( x ) = \frac { x ^ { 2 } } { 6 x + 1 }f(x)=6x+1x2​

Use the Remainder Theorem and synthetic division to find the function value. Verify your answers using another method. h(x)=x3−6x2−7x+5,h(−6)h ( x ) = x ^ { 3 } - 6 x ^ { 2 } - 7 x + 5 , \quad h ( - 6 )h(x)=x3−6x2−7x+5,h(−6)

Select the graph of. fff f(x)=x3+x2−4x−4f ( x ) = x ^ { 3 } + x ^ { 2 } - 4 x - 4f(x)=x3+x2−4x−4

Write the quadratic function f(x)=−x2+2x+2f ( x ) = - x ^ { 2 } + 2 x + 2f(x)=−x2+2x+2 in standard form.

Use Descartes's Rule of Signs to determine the possible numbers of positive and neg- ative zeros of the function. 5x5−10x5 x ^ { 5 } - 10 x5x5−10x

Find the domain of xxx in the expression. Use a graphing utility to verify your result. 25−16x2\sqrt { 25 - 16 x ^ { 2 } }25−16x2​

Perform the addition or subtraction and write the result in standard form. (9+i)+(2−6i)( 9 + i ) + ( 2 - 6 i )(9+i)+(2−6i)

Perform the addition or subtraction and write the result in standard form. 17i−(18−8i)17 i - ( 18 - 8 i )17i−(18−8i)

Perform the operation and write the result in standard form. 51+i−61−i\frac { 5 } { 1 + i } - \frac { 6 } { 1 - i }1+i5​−1−i6​

Simplify the complex number and write it in standard form. (−i)5( - i ) ^ { 5 }(−i)5

Write the complex number in standard form. (3+−6)(6−−6)( 3 + \sqrt { - 6 } ) ( 6 - \sqrt { - 6 } )(3+−6​)(6−−6​)

Use long division to divide. (x4+4x2+5)÷(x2+x+4)\left( x ^ { 4 } + 4 x ^ { 2 } + 5 \right) \div \left( x ^ { 2 } + x + 4 \right)(x4+4x2+5)÷(x2+x+4)

Simplify the rational expression, x4+x3−21x2−41x−20x2+2x+1\frac { x ^ { 4 } + x ^ { 3 } - 21 x ^ { 2 } - 41 x - 20 } { x ^ { 2 } + 2 x + 1 }x2+2x+1x4+x3−21x2−41x−20​ ,by using long division or synthetic division.

Select the graph of the function y=−3x2y = - 3 x ^ { 2 }y=−3x2 . Compare the graph of this function with the graph of y=x2y = x ^ { 2 }y=x2

Find all the zeros of the function and write the polynomial as a product of linear fac- tors. y4−2401y ^ { 4 } - 2401y4−2401

Determine the equations of any horizontal and vertical asymptotes of f(x)=x2+8x+15x3+5x2−x−5f ( x ) = \frac { x ^ { 2 } + 8 x + 15 } { x ^ { 3 } + 5 x ^ { 2 } - x - 5 }f(x)=x3+5x2−x−5x2+8x+15​

Solve the equation and write complex solutions in standard form. x2−6x+58=0x ^ { 2 } - 6 x + 58 = 0x2−6x+58=0

Select the graph of the function and determine the zeros of the polynomial. f(x)=x3−4x2f ( x ) = x ^ { 3 } - 4 x ^ { 2 }f(x)=x3−4x2

Functions and Their Graphs

Exponential and Logarithmic Functions

Trigonometry

Analytic Trigonometry

Topics in Analytic Geometry

Filters

Find all the real zeros of the polynomial function. $f ( x ) = x ^ { 5 } - 10 x ^ { 3 } + 25 x$

Select the correct graph of the following function. $f ( x ) = \frac { x ^ { 2 } } { 6 x + 1 }$

Use the Remainder Theorem and synthetic division to find the function value. Verify your answers using another method. $h ( x ) = x ^ { 3 } - 6 x ^ { 2 } - 7 x + 5 , \quad h ( - 6 )$

Select the graph of. $f$ $f ( x ) = x ^ { 3 } + x ^ { 2 } - 4 x - 4$

Write the quadratic function $f ( x ) = - x ^ { 2 } + 2 x + 2$ in standard form.

Use Descartes's Rule of Signs to determine the possible numbers of positive and neg- ative zeros of the function. $5 x ^ { 5 } - 10 x$

Find the domain of $x$ in the expression. Use a graphing utility to verify your result. $\sqrt { 25 - 16 x ^ { 2 } }$

Perform the addition or subtraction and write the result in standard form. $( 9 + i ) + ( 2 - 6 i )$

Perform the addition or subtraction and write the result in standard form. $17 i - ( 18 - 8 i )$

Perform the operation and write the result in standard form. $\frac { 5 } { 1 + i } - \frac { 6 } { 1 - i }$

Simplify the complex number and write it in standard form. $( - i ) ^ { 5 }$

Write the complex number in standard form. $( 3 + \sqrt { - 6 } ) ( 6 - \sqrt { - 6 } )$

Use long division to divide. $\left( x ^ { 4 } + 4 x ^ { 2 } + 5 \right) \div \left( x ^ { 2 } + x + 4 \right)$

Simplify the rational expression, $\frac { x ^ { 4 } + x ^ { 3 } - 21 x ^ { 2 } - 41 x - 20 } { x ^ { 2 } + 2 x + 1 }$ ,by using long division or synthetic division.

Select the graph of the function $y = - 3 x ^ { 2 }$ . Compare the graph of this function with the graph of $y = x ^ { 2 }$

Find all the zeros of the function and write the polynomial as a product of linear fac- tors. $y ^ { 4 } - 2401$

Determine the equations of any horizontal and vertical asymptotes of $f ( x ) = \frac { x ^ { 2 } + 8 x + 15 } { x ^ { 3 } + 5 x ^ { 2 } - x - 5 }$

Solve the equation and write complex solutions in standard form. $x ^ { 2 } - 6 x + 58 = 0$

Select the graph of the function and determine the zeros of the polynomial. $f ( x ) = x ^ { 3 } - 4 x ^ { 2 }$