Question 1

Use the remainder theorem and synthetic division to find f(k).
-$k = 4 - 3 i ; f ( x ) = x ^ { 2 } + 2 x + 5$

Accepted Answer

A)  $20 - 18 \mathrm { i }$ 
B)  $- 10$ 
C)  $20 - 30 \mathrm { i }$ 
D)  $15 - 18 \mathrm { i }$

Question 2

Use the boundedness theorem to determine whether the polynomial function satisfies the given condition.
-The polynomial $f ( x ) = x ^ { 5 } + 10 x ^ { 4 } - 25 x ^ { 3 } - 10 x ^ { 2 } + 24 x$ has no real zero less than $- 8$.

Accepted Answer

A)  Yes, the boundedness theorem shows that the polynomial has no real zero less than $- 8$. 
B)  No, the boundedness theorem does not show that the polynomial has no real zero less than $- 8$

Question 3

Translate the given formula to an English phrase using the word "varies".
-$$\text { What type of variation is suggested by the graph? }$$

Accepted Answer

A)  y varies directly as the square of x 
B)  y varies directly as x 
C)  y varies inversely as x 
D)  y varies inversely as the square of x

Question 4

Use synthetic division to divide f(x) by x - k for the given value of k. Then express f(x) in the form f(x) = (x - k)q(x) + r for
the given value of k.
-$$f ( x ) = 2 x ^ { 4 } - x ^ { 3 } - 15 x ^ { 2 } + 3 x ; k = - 3$$

Accepted Answer

A)  $f ( x ) = ( x + 3 ) \left( 2 x ^ { 3 } - 7 x ^ { 2 } + 6 x - 15 \right) + 45$ 
B)  $f ( x ) = ( x - 3 ) \left( 2 x ^ { 3 } - 7 x ^ { 2 } + 6 x - 15 \right) + 45$ 
C)  $f ( x ) = ( x - 3 ) \left( 2 x ^ { 3 } - 5 x ^ { 2 } + 3 \right) + 9$ 
D)  $f ( x ) = ( x + 3 ) \left( 2 x ^ { 3 } + 5 x ^ { 2 } + 3 \right) + 9$

Question 5

Give the equation of any oblique asymptotes.
-$$f ( x ) = \frac { x ^ { 2 } + 4 x - 7 } { x - 5 }$$

Accepted Answer

A)  $x = y + 9$ 
B)  $y = x + 9$ 
C)  $y = x - 1$ 
D)  none

Question 6

Use synthetic division to decide whether the given number k is a zero of the given polynomial function.
-$k = 2 + i ; f ( x ) = x ^ { 3 } + 5 x ^ { 2 } - 12 x + 14$

Accepted Answer

A)  Yes 
B)  No

Question 7

Sketch the graph of the parabola.
-$$y = - x ^ { 2 } + 1$$

Accepted Answer

A) 
 
B) 
 
C) 
 
D)

Question 8

Translate the given formula to an English phrase using the word "varies".
-If y varies inversely as the square of x, and x is tripled, how is y changed?

Accepted Answer

A)  y is one-ninth as large as before 
B)  y is one-third as large as before 
C)  y is nine times as large as before 
D)  y is three times as large as before

Question 9

Find a polynomial function f(x) of least possible degree having the graph shown.
-

Accepted Answer

A)  $f ( x ) = - 6 ( x + 1 ) ^ { 2 } ( x - 1 ) ^ { 2 }$ 
B)  $f ( x ) = 6 ( x + 1 ) ^ { 2 } ( x - 1 ) ^ { 2 }$ 
C)  $f ( x ) = - 6 ( x + 1 ) ( x - 1 )$ 
D)  $f ( x ) = 6 ( x + 1 ) ( x - 1 )$

Question 10

Find the domain and range of the function.
-$$y = ( x + 1 ) ^ { 2 } + 5$$

Accepted Answer

A)  Domain: $( - \infty , \infty )$; Range: $[ 5 , \infty )$ 
B)  Domain: $[ 5 , \infty )$; Range: $[ - \infty , 5 )$ 
C)  Domain: $( - \infty , \infty )$; Range: $[ - 1 , \infty )$ 
D)  Domain: $[ - 1 , \infty )$; Range: $[ - \infty , 1 )$

Question 11

Find the correct end behavior diagram for the given polynomial function.
-$$P ( x ) = - 5 x ^ { 3 } + 9 x ^ { 2 } + 4 x - 5$$

Accepted Answer

A)  
 
B)  
 
C) 
 
D)

Question 12

Determine which of the rational functions given below has the following feature(s).
-The y-intercept is $( 0,4 )$

Accepted Answer

A)  $f ( x ) = \frac { x - 8 } { x - 2 }$ 
B)  $f ( x ) = \frac { 5 x - 1 } { x + 4 }$ 
C)  $f ( x ) = \frac { x + 8 } { x - 2 }$ 
D)  $f ( x ) = \frac { x + 4 } { x - 2 }$

Question 13

Round your answer to two decimal places.
-The weight that a horizontal beam can support varies inversely as the length of the beam. Suppose that a 5-m beam can support 760 kg. How many kilograms can a 1-m beam support?

Accepted Answer

A)  0.0066 kg 
B)  0.0003 kg 
C)  3800 kg 
D)  152 kg

Question 14

Find the equation that the given graph represents.
-

Accepted Answer

A)  $P ( x ) = x ^ { 3 } + x ^ { 2 } + x + 5$ 
B)  $P ( x ) = - x ^ { 3 } + 4 x ^ { 2 } + x - 5$ 
C)  $P ( x ) = x ^ { 5 } + 4 x ^ { 3 } - x ^ { 2 } + 3 x - 5$ 
D)  $P ( x ) = x ^ { 3 } + 4 x ^ { 2 } - x - 5$

Question 15

Provide an appropriate response.
-Why does a function defined by a polynomial of degree five with real coefficients have 
either 1, 3, or 5 real zeros counting multiplicities?

Accepted Answer

The answer of Provide an appropriate response.
-Why does a function...

Question 16

Use synthetic division to decide whether the given number k is a zero of the given polynomial function.
-$$k = 2 ; f ( x ) = 7 x ^ { 3 } + x ^ { 2 } + 2 x - 3$$

Accepted Answer

A)  Yes 
B)  No

Question 17

Tell whether a linear model or a quadratic model is appropriate for the data. If linear, tell whether the slope should be
positive or negative. If quadratic, decide whether the leading coefficient a of $x ^ { 2 }$ should be positive or negative.
-

Accepted Answer

A)  quadratic; negative 
B)  linear; positive 
C)  quadratic; positive 
D)  linear; negative

Question 18

Use a graphing calculator to approximate the real zeros. Give each zero as a decimal to the nearest tenth.
-$$f ( x ) = 4 x ^ { 3 } + x ^ { 2 } - 12 x - 3$$

Accepted Answer

A)  $- 4,1.8 , - 1.8$ 
B)  $0.3,1.7 , - 1.7$ 
C)  $- 0.3,1.7 , - 1.7$ 
D)  $4,3 , - 3$

Question 19

Graph the polynomial function. Factor first if the expression is not in factored form.
-$$f ( x ) = ( 2 x + 1 ) ( x - 1 ) ( x + 2 )$$

Accepted Answer

A) 
 
B) 
 
C) 
 
D)

Question 20

Find the domain and range of the function.
-$$f ( x ) = x ^ { 2 } + 4$$

Accepted Answer

A)  Domain: $( - \infty , \infty )$; range: $( - 4 , \infty )$ 
B)  Domain: $( - 4 , \infty )$; range: $( - \infty , \infty )$ 
C)  Domain: $( - \infty , \infty )$; range: $[ 4 , \infty )$ 
D)  Domain: $( 4 , \infty )$; range: $( - \infty , \infty )$

Exam 4: Polynomial and Rational Functions

Use the remainder theorem and synthetic division to find f(k) . - $k = 4 - 3 i ; f ( x ) = x ^ { 2 } + 2 x + 5$

Use the boundedness theorem to determine whether the polynomial function satisfies the given condition. -The polynomial $f ( x ) = x ^ { 5 } + 10 x ^ { 4 } - 25 x ^ { 3 } - 10 x ^ { 2 } + 24 x$ has no real zero less than $- 8$ .

Use synthetic division to divide f(x) by x - k for the given value of k. Then express f(x) in the form f(x) = (x - k) q(x) + r for the given value of k. - $f ( x ) = 2 x ^ { 4 } - x ^ { 3 } - 15 x ^ { 2 } + 3 x ; k = - 3$

Give the equation of any oblique asymptotes. - $f ( x ) = \frac { x ^ { 2 } + 4 x - 7 } { x - 5 }$

Use synthetic division to decide whether the given number k is a zero of the given polynomial function. - $k = 2 + i ; f ( x ) = x ^ { 3 } + 5 x ^ { 2 } - 12 x + 14$

Sketch the graph of the parabola. - $y = - x ^ { 2 } + 1$ $Sketch the graph of the parabola. - y = - x ^ { 2 } + 1 A) B) C) D)$

Translate the given formula to an English phrase using the word "varies". -If y varies inversely as the square of x, and x is tripled, how is y changed?

Find the domain and range of the function. - $y = ( x + 1 ) ^ { 2 } + 5$

Find the correct end behavior diagram for the given polynomial function. - $P ( x ) = - 5 x ^ { 3 } + 9 x ^ { 2 } + 4 x - 5$

Determine which of the rational functions given below has the following feature(s) . -The y-intercept is $( 0,4 )$

Round your answer to two decimal places. -The weight that a horizontal beam can support varies inversely as the length of the beam. Suppose that a 5-m beam can support 760 kg. How many kilograms can a 1-m beam support?

Provide an appropriate response. -Why does a function defined by a polynomial of degree five with real coefficients have either 1, 3, or 5 real zeros counting multiplicities?

Use synthetic division to decide whether the given number k is a zero of the given polynomial function. - $k = 2 ; f ( x ) = 7 x ^ { 3 } + x ^ { 2 } + 2 x - 3$

Use a graphing calculator to approximate the real zeros. Give each zero as a decimal to the nearest tenth. - $f ( x ) = 4 x ^ { 3 } + x ^ { 2 } - 12 x - 3$

Graph the polynomial function. Factor first if the expression is not in factored form. - $f ( x ) = ( 2 x + 1 ) ( x - 1 ) ( x + 2 )$

Find the domain and range of the function. - $f ( x ) = x ^ { 2 } + 4$

Exam 4: Polynomial and Rational Functions

Use the remainder theorem and synthetic division to find f(k) . - k=4−3i;f(x) =x2+2x+5k = 4 - 3 i ; f ( x ) = x ^ { 2 } + 2 x + 5k=4−3i;f(x) =x2+2x+5

Use the boundedness theorem to determine whether the polynomial function satisfies the given condition. -The polynomial f(x) =x5+10x4−25x3−10x2+24xf ( x ) = x ^ { 5 } + 10 x ^ { 4 } - 25 x ^ { 3 } - 10 x ^ { 2 } + 24 xf(x) =x5+10x4−25x3−10x2+24x has no real zero less than −8- 8−8 .

Translate the given formula to an English phrase using the word "varies". - What type of variation is suggested by the graph? \text { What type of variation is suggested by the graph? } What type of variation is suggested by the graph?

Use synthetic division to divide f(x) by x - k for the given value of k. Then express f(x) in the form f(x) = (x - k) q(x) + r for the given value of k. - f(x) =2x4−x3−15x2+3x;k=−3f ( x ) = 2 x ^ { 4 } - x ^ { 3 } - 15 x ^ { 2 } + 3 x ; k = - 3f(x) =2x4−x3−15x2+3x;k=−3

Give the equation of any oblique asymptotes. - f(x) =x2+4x−7x−5f ( x ) = \frac { x ^ { 2 } + 4 x - 7 } { x - 5 }f(x) =x−5x2+4x−7​

Use synthetic division to decide whether the given number k is a zero of the given polynomial function. - k=2+i;f(x) =x3+5x2−12x+14k = 2 + i ; f ( x ) = x ^ { 3 } + 5 x ^ { 2 } - 12 x + 14k=2+i;f(x) =x3+5x2−12x+14

Sketch the graph of the parabola. - y=−x2+1y = - x ^ { 2 } + 1y=−x2+1

Translate the given formula to an English phrase using the word "varies". -If y varies inversely as the square of x, and x is tripled, how is y changed?

Find a polynomial function f(x) of least possible degree having the graph shown. -

Find the domain and range of the function. - y=(x+1) 2+5y = ( x + 1 ) ^ { 2 } + 5y=(x+1) 2+5

Find the correct end behavior diagram for the given polynomial function. - P(x) =−5x3+9x2+4x−5P ( x ) = - 5 x ^ { 3 } + 9 x ^ { 2 } + 4 x - 5P(x) =−5x3+9x2+4x−5

Determine which of the rational functions given below has the following feature(s) . -The y-intercept is (0,4) ( 0,4 ) (0,4)