Question 1

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not.
-$$f ( x ) = \frac { 2 - x ^ { 2 } } { 5 }$$

Accepted Answer

A)  No; x is a negative term 
B)  Yes; degree 2 
C)  Yes; degree 1 
D)  No; it is a ratio 
A)  No; x is a negative term 
B)  Yes; degree 2 
C)  Yes; degree 1 
D)  No; it is a ratio

Question 2

$f ( x ) = \frac { 1 } { 2 } ( x + 2 ) ^ { 4 } + 2$

Accepted Answer

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

Question 3

Use Descartes' Rule of Signs and the Rational Zeros Theorem to find all the real zeros of the polynomial function. Use the
zeros to factor f over the real numbers.
-$$f ( x ) = x ^ { 4 } - 15 x ^ { 2 } - 16$$

Accepted Answer

A)  $- 4,4 ; f ( x ) = ( x - 4 ) ( x + 4 ) \left( x ^ { 2 } + 1 \right)$ 
B)  $4 ; f ( x ) = ( x - 4 ) ^ { 2 } \left( x ^ { 2 } + 1 \right)$ 
C)  $- 1,1 ; f ( x ) = ( x - 1 ) ( x + 1 ) \left( x ^ { 2 } + 16 \right)$ 
D)  $- 4 , - 1,4,1 ; f ( x ) = ( x - 4 ) ( x + 4 ) ( x - 1 ) ( x + 1 )$ 
A)  $- 4,4 ; f ( x ) = ( x - 4 ) ( x + 4 ) \left( x ^ { 2 } + 1 \right)$ 
B)  $4 ; f ( x ) = ( x - 4 ) ^ { 2 } \left( x ^ { 2 } + 1 \right)$ 
C)  $- 1,1 ; f ( x ) = ( x - 1 ) ( x + 1 ) \left( x ^ { 2 } + 16 \right)$ 
D)  $- 4 , - 1,4,1 ; f ( x ) = ( x - 4 ) ( x + 4 ) ( x - 1 ) ( x + 1 )$

Question 4

Solve the problem.
-Which of the following polynomial functions might have the graph shown in the illustration below?

Accepted Answer

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

Question 5

Solve the problem.
-The amount of water (in gallons) in a leaky bathtub is given in the table below. Using a graphing utility, fit the data to a third degree polynomial (or a cubic). Then approximate the time at which there is maximum amount
Of water in the tub, and estimate the time when the water runs out of the tub. Express all your answers rounded
To two decimal places. $$\begin{array} { c | c c c c c c c c } 
\mathrm { t } \text { (in minutes) } & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \
\hline \mathrm { V } \text { (in gallons) } & 20 & 26 & 45 & 63 & 86 & 94 & 90 & 67
\end{array}$$

Accepted Answer

A)  maximum amount of water after 8.23 minutes; water runs out after 19.73 minutes 
B)  maximum amount of water after 5.31 minutes; water runs out after 8.23 minutes 
C)  maximum amount of water after 5.37 minutes; water runs out after 11.06 minutes 
D)  maximum amount of water after 5.31 minutes; water never runs out 
A)  maximum amount of water after 8.23 minutes; water runs out after 19.73 minutes 
B)  maximum amount of water after 5.31 minutes; water runs out after 8.23 minutes 
C)  maximum amount of water after 5.37 minutes; water runs out after 11.06 minutes 
D)  maximum amount of water after 5.31 minutes; water never runs out

Question 6

Solve the equation in the real number system.
-$$x ^ { 3 } + 8 x ^ { 2 } - 18 x + 20 = 0$$

Accepted Answer

A)  {1} 
B)  {10} 
C)  {-10, 10} 
D)  {-10} 
A)  {1} 
B)  {10} 
C)  {-10, 10} 
D)  {-10}

Question 7

Determine the maximum number of turning points of f.
-$$f ( x ) = ( x - 3 ) ^ { 2 } ( x + 5 ) ^ { 2 }$$

Accepted Answer

A)  above the $x$-axis: $( - \infty , - 5 ) , ( 3 , \infty )$
below the x-axis: $( - 5,3 )$ 
B)  above the x-axis: $( - 5,3 )$
below the $x$-axis: $( - \infty , - 5 ) , ( 3 , \infty )$ 
C)  above the $x$-axis: $( - \infty , - 5 ) , ( - 5,3 ) , ( 3 , \infty )$
below the $x$-axis: no intervals 
D)  above the $x$-axis: no intervals
below the $x$-axis: $( - \infty , - 5 ) , ( - 5,3 ) , ( 3 , \infty )$ 
A)  above the $x$-axis: $( - \infty , - 5 ) , ( 3 , \infty )$
below the x-axis: $( - 5,3 )$ 
B)  above the x-axis: $( - 5,3 )$
below the $x$-axis: $( - \infty , - 5 ) , ( 3 , \infty )$ 
C)  above the $x$-axis: $( - \infty , - 5 ) , ( - 5,3 ) , ( 3 , \infty )$
below the $x$-axis: no intervals 
D)  above the $x$-axis: no intervals
below the $x$-axis: $( - \infty , - 5 ) , ( - 5,3 ) , ( 3 , \infty )$

Question 8

Find the x- and y-intercepts of f.
-$$f ( x ) = ( x + 13 ) ^ { 2 }$$

Accepted Answer

A)  x-intercept: 13; y-intercept: 169 
B)  x-intercept: -13; y-intercept: 0 
C)  x-intercept: -13; y-intercept: 169 
D)  x-intercept: 13; y-intercept: 0 
A)  x-intercept: 13; y-intercept: 169 
B)  x-intercept: -13; y-intercept: 0 
C)  x-intercept: -13; y-intercept: 169 
D)  x-intercept: 13; y-intercept: 0

Question 9

Find a bound on the real zeros of the polynomial function.
-f(x) = 13x3 - x2 + 0.3x - 0.06

Accepted Answer

A)  -2 and 2 
B)  -1.36 and 1.36 
C)  -13 and 13 
D)  -1 and 1 
A)  -2 and 2 
B)  -1.36 and 1.36 
C)  -13 and 13 
D)  -1 and 1

Question 10

Use the Intermediate Value Theorem to determine whether the polynomial function has a zero in the given interval.
-$$f ( x ) = 2 x ^ { 4 } - 17 x ^ { 3 } + 58 x ^ { 2 } - 77 x + 26 ; \text { zero: } 3 + 2 i$$

Accepted Answer

A)  $2 - 3 i , 2 , - \frac { 1 } { 2 }$ 
B)  $3 - 2 i , 2 , \frac { 1 } { 2 }$ 
C)  $2 - 3 \mathrm { i } , - 2 , - \frac { 1 } { 2 }$ 
D)  $3 - 2 \mathrm { i } , - 2 , \frac { 1 } { 2 }$ 
A)  $2 - 3 i , 2 , - \frac { 1 } { 2 }$ 
B)  $3 - 2 i , 2 , \frac { 1 } { 2 }$ 
C)  $2 - 3 \mathrm { i } , - 2 , - \frac { 1 } { 2 }$ 
D)  $3 - 2 \mathrm { i } , - 2 , \frac { 1 } { 2 }$

Question 11

Find the domain of the rational function.
-$$h ( x ) = \frac { 2 x } { ( x - 7 ) ( x - 4 ) }$$

Accepted Answer

A)  $\{ x \mid x \neq 7 , x \neq 4 , x \neq - 2 \}$ 
B)  all real numbers 
C)  $\{ x \mid x \neq 7 , x \neq 4 \}$ 
D)  $\{ x \mid x \neq - 7 , x \neq - 4 \}$ 
A)  $\{ x \mid x \neq 7 , x \neq 4 , x \neq - 2 \}$ 
B)  all real numbers 
C)  $\{ x \mid x \neq 7 , x \neq 4 \}$ 
D)  $\{ x \mid x \neq - 7 , x \neq - 4 \}$

Question 12

Find the x- and y-intercepts of f.
-f(x) = 4x2 (x - 4)5

Accepted Answer

A)  x-intercepts: 0, -4; y-intercept: 0 
B)  x-intercepts: 0, 4; y-intercept: 4 
C)  x-intercepts: 0, -4; y-intercept: 4 
D)  x-intercepts: 0, 4; y-intercept: 0 
A)  x-intercepts: 0, -4; y-intercept: 0 
B)  x-intercepts: 0, 4; y-intercept: 4 
C)  x-intercepts: 0, -4; y-intercept: 4 
D)  x-intercepts: 0, 4; y-intercept: 0

Question 13

f(x) = -2(x - 5)4 + 3

Accepted Answer

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

Question 14

Use the Intermediate Value Theorem to determine whether the polynomial function has a zero in the given interval.
-$f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 9 x + 13$; zero: $2 + 3 i$

Accepted Answer

A)  $2 - 3 i$, 1 
B)  $2 - 3 \mathrm { i } , - 1$ 
C)  $3 - 2 i , - 1$ 
D)  $3 - 2 \mathrm { i } , 1$ 
A)  $2 - 3 i$, 1 
B)  $2 - 3 \mathrm { i } , - 1$ 
C)  $3 - 2 i , - 1$ 
D)  $3 - 2 \mathrm { i } , 1$

Question 15

Find the x- and y-intercepts of f.
-f(x) = (x - 3)(x - 1)

Accepted Answer

A)  x-intercepts: 3, 1; y-intercept: -4 
B)  x-intercepts: -3, -1; y-intercept: -4 
C)  x-intercepts: 3, 1; y-intercept: 3 
D)  x-intercepts: -3, -1; y-intercept: 3 
A)  x-intercepts: 3, 1; y-intercept: -4 
B)  x-intercepts: -3, -1; y-intercept: -4 
C)  x-intercepts: 3, 1; y-intercept: 3 
D)  x-intercepts: -3, -1; y-intercept: 3

Question 16

Solve the equation in the real number system.
-$3 x ^ { 4 } - 29 x ^ { 3 } + 111 x ^ { 2 } - 179 x + 78 = 0$

Accepted Answer

A)  $\left\{ - 3 , - \frac { 2 } { 3 } \right\}$ 
B)  $\left\{ - 3 , \frac { 2 } { 3 } \right\}$ 
C)  $\left\{ 3 , - \frac { 2 } { 3 } \right\}$ 
D)  $\left\{ 3 , \frac { 2 } { 3 } \right\}$ 
A)  $\left\{ - 3 , - \frac { 2 } { 3 } \right\}$ 
B)  $\left\{ - 3 , \frac { 2 } { 3 } \right\}$ 
C)  $\left\{ 3 , - \frac { 2 } { 3 } \right\}$ 
D)  $\left\{ 3 , \frac { 2 } { 3 } \right\}$

Question 17

Use the Intermediate Value Theorem to determine whether the polynomial function has a zero in the given interval.
-f(x) = 2x4 + 8x3 + 4x - 2; [-5, -4]

Accepted Answer

A)  f(-5) = -228 and f(-4) = 18; yes 
B)  f(-5) = 228 and f(-4) = -18; yes 
C)  f(-5) = 228 and f(-4) = 18; no 
D)  f(-5) = -228 and f(-4) = -18; no 
A)  f(-5) = -228 and f(-4) = 18; yes 
B)  f(-5) = 228 and f(-4) = -18; yes 
C)  f(-5) = 228 and f(-4) = 18; no 
D)  f(-5) = -228 and f(-4) = -18; no

Question 18

Find a bound on the real zeros of the polynomial function.
-f(x) = x5 + 4x4 + 3x3 + 2x2 - 3x + 4

Accepted Answer

A)  -5 and 5 
B)  - 4 and 4 
C)  -16 and 16 
D)  -11 and 11 
A)  -5 and 5 
B)  - 4 and 4 
C)  -16 and 16 
D)  -11 and 11

Question 19

f(x) = (x + 2)5 + 4

Accepted Answer

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

Question 20

Find the x- and y-intercepts of f.
-f(x) = x2 (x - 1)(x - 3)

Accepted Answer

A)  x-intercepts: 0, -1, -3; y-intercept: 3 
B)  x-intercepts: 0, 1, 3; y-intercept: 3 
C)  x-intercepts: 0, 1, 3; y-intercept: 0 
D)  x-intercepts: 0, -1, -3; y-intercept: 0 
A)  x-intercepts: 0, -1, -3; y-intercept: 3 
B)  x-intercepts: 0, 1, 3; y-intercept: 3 
C)  x-intercepts: 0, 1, 3; y-intercept: 0 
D)  x-intercepts: 0, -1, -3; y-intercept: 0

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not. - $f ( x ) = \frac { 2 - x ^ { 2 } } { 5 }$

$f ( x ) = \frac { 1 } { 2 } ( x + 2 ) ^ { 4 } + 2$ $f ( x ) = \frac { 1 } { 2 } ( x + 2 ) ^ { 4 } + 2$

Use Descartes' Rule of Signs and the Rational Zeros Theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers. - $f ( x ) = x ^ { 4 } - 15 x ^ { 2 } - 16$

Solve the problem. -Which of the following polynomial functions might have the graph shown in the illustration below?

Solve the equation in the real number system. - $x ^ { 3 } + 8 x ^ { 2 } - 18 x + 20 = 0$

Determine the maximum number of turning points of f. - $f ( x ) = ( x - 3 ) ^ { 2 } ( x + 5 ) ^ { 2 }$

Find the x- and y-intercepts of f. - $f ( x ) = ( x + 13 ) ^ { 2 }$

Find a bound on the real zeros of the polynomial function. -f(x) = 13x³ - x² + 0.3x - 0.06

Use the Intermediate Value Theorem to determine whether the polynomial function has a zero in the given interval. - $f ( x ) = 2 x ^ { 4 } - 17 x ^ { 3 } + 58 x ^ { 2 } - 77 x + 26 ; \text { zero: } 3 + 2 i$

Find the domain of the rational function. - $h ( x ) = \frac { 2 x } { ( x - 7 ) ( x - 4 ) }$

Find the x- and y-intercepts of f. -f(x) = 4x² (x - 4)5

f(x) = -2(x - 5)⁴ + 3

Use the Intermediate Value Theorem to determine whether the polynomial function has a zero in the given interval. - $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 9 x + 13$ ; zero: $2 + 3 i$

Find the x- and y-intercepts of f. -f(x) = (x - 3)(x - 1)

Solve the equation in the real number system. - $3 x ^ { 4 } - 29 x ^ { 3 } + 111 x ^ { 2 } - 179 x + 78 = 0$

Use the Intermediate Value Theorem to determine whether the polynomial function has a zero in the given interval. -f(x) = 2x⁴ + 8x³ + 4x - 2; [-5, -4]

Find a bound on the real zeros of the polynomial function. -f(x) = x⁵ + 4x⁴ + 3x³ + 2x² - 3x + 4

f(x) = (x + 2)⁵ + 4

Find the x- and y-intercepts of f. -f(x) = x² (x - 1)(x - 3)

Functions and Their Graphs

Linear and Quadratic Functions

Conics

Systems of Equations and Inequalities

Sequences; Induction; the Binomial Theorem

Counting and Probability

Filters

Exam 3: Polynomial and Rational Functions

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not. - f(x)=2−x25f ( x ) = \frac { 2 - x ^ { 2 } } { 5 }f(x)=52−x2​

f(x)=12(x+2)4+2f ( x ) = \frac { 1 } { 2 } ( x + 2 ) ^ { 4 } + 2f(x)=21​(x+2)4+2

Use Descartes' Rule of Signs and the Rational Zeros Theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers. - f(x)=x4−15x2−16f ( x ) = x ^ { 4 } - 15 x ^ { 2 } - 16f(x)=x4−15x2−16

Solve the problem. -Which of the following polynomial functions might have the graph shown in the illustration below?

Solve the equation in the real number system. - x3+8x2−18x+20=0x ^ { 3 } + 8 x ^ { 2 } - 18 x + 20 = 0x3+8x2−18x+20=0

Determine the maximum number of turning points of f. - f(x)=(x−3)2(x+5)2f ( x ) = ( x - 3 ) ^ { 2 } ( x + 5 ) ^ { 2 }f(x)=(x−3)2(x+5)2

Find the x- and y-intercepts of f. - f(x)=(x+13)2f ( x ) = ( x + 13 ) ^ { 2 }f(x)=(x+13)2

Find a bound on the real zeros of the polynomial function. -f(x) = 13x3 - x2 + 0.3x - 0.06

Use the Intermediate Value Theorem to determine whether the polynomial function has a zero in the given interval. - f(x)=2x4−17x3+58x2−77x+26; zero: 3+2if ( x ) = 2 x ^ { 4 } - 17 x ^ { 3 } + 58 x ^ { 2 } - 77 x + 26 ; \text { zero: } 3 + 2 if(x)=2x4−17x3+58x2−77x+26; zero: 3+2i

Find the domain of the rational function. - h(x)=2x(x−7)(x−4)h ( x ) = \frac { 2 x } { ( x - 7 ) ( x - 4 ) }h(x)=(x−7)(x−4)2x​

Find the x- and y-intercepts of f. -f(x) = 4x2 (x - 4)5

f(x) = -2(x - 5)4 + 3

Use the Intermediate Value Theorem to determine whether the polynomial function has a zero in the given interval. - f(x)=x3−3x2+9x+13f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 9 x + 13f(x)=x3−3x2+9x+13 ; zero: 2+3i2 + 3 i2+3i

Find the x- and y-intercepts of f. -f(x) = (x - 3)(x - 1)

Solve the equation in the real number system. - 3x4−29x3+111x2−179x+78=03 x ^ { 4 } - 29 x ^ { 3 } + 111 x ^ { 2 } - 179 x + 78 = 03x4−29x3+111x2−179x+78=0

Use the Intermediate Value Theorem to determine whether the polynomial function has a zero in the given interval. -f(x) = 2x4 + 8x3 + 4x - 2; [-5, -4]

Find a bound on the real zeros of the polynomial function. -f(x) = x5 + 4x4 + 3x3 + 2x2 - 3x + 4

f(x) = (x + 2)5 + 4

Find the x- and y-intercepts of f. -f(x) = x2 (x - 1)(x - 3)

Functions and Their Graphs

Linear and Quadratic Functions

Conics

Systems of Equations and Inequalities

Sequences; Induction; the Binomial Theorem

Counting and Probability

Filters

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not. - $f ( x ) = \frac { 2 - x ^ { 2 } } { 5 }$

$f ( x ) = \frac { 1 } { 2 } ( x + 2 ) ^ { 4 } + 2$ $f ( x ) = \frac { 1 } { 2 } ( x + 2 ) ^ { 4 } + 2$

Use Descartes' Rule of Signs and the Rational Zeros Theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers. - $f ( x ) = x ^ { 4 } - 15 x ^ { 2 } - 16$

Solve the equation in the real number system. - $x ^ { 3 } + 8 x ^ { 2 } - 18 x + 20 = 0$

Determine the maximum number of turning points of f. - $f ( x ) = ( x - 3 ) ^ { 2 } ( x + 5 ) ^ { 2 }$

Find the x- and y-intercepts of f. - $f ( x ) = ( x + 13 ) ^ { 2 }$

Find a bound on the real zeros of the polynomial function. -f(x) = 13x³ - x² + 0.3x - 0.06

Use the Intermediate Value Theorem to determine whether the polynomial function has a zero in the given interval. - $f ( x ) = 2 x ^ { 4 } - 17 x ^ { 3 } + 58 x ^ { 2 } - 77 x + 26 ; \text { zero: } 3 + 2 i$

Find the domain of the rational function. - $h ( x ) = \frac { 2 x } { ( x - 7 ) ( x - 4 ) }$

Find the x- and y-intercepts of f. -f(x) = 4x² (x - 4)5

f(x) = -2(x - 5)⁴ + 3

Use the Intermediate Value Theorem to determine whether the polynomial function has a zero in the given interval. - $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 9 x + 13$ ; zero: $2 + 3 i$

Solve the equation in the real number system. - $3 x ^ { 4 } - 29 x ^ { 3 } + 111 x ^ { 2 } - 179 x + 78 = 0$

Use the Intermediate Value Theorem to determine whether the polynomial function has a zero in the given interval. -f(x) = 2x⁴ + 8x³ + 4x - 2; [-5, -4]

Find a bound on the real zeros of the polynomial function. -f(x) = x⁵ + 4x⁴ + 3x³ + 2x² - 3x + 4

f(x) = (x + 2)⁵ + 4

Find the x- and y-intercepts of f. -f(x) = x² (x - 1)(x - 3)