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)  $15 - 18 \mathrm { i }$ 
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 $- 8$ 
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 
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)  $f ( x ) = ( x + 3 ) \left( 2 x ^ { 3 } + 5 x ^ { 2 } + 3 \right) + 9$ 
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 
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 
A)  Yes 
B)  No

Question 7

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

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
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 
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)  $f ( x ) = 6 ( x + 1 ) ( x - 1 )$ 
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: $[ - \infty , 1 )$ 
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) 
  
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)  $f ( x ) = \frac { x + 4 } { x - 2 }$ 
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 
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)  $P ( x ) = x ^ { 3 } + 4 x ^ { 2 } - x - 5$ 
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

By the Fundamental Theorem of Algebra an

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 
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 
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)  $4,3 , - 3$ 
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) 
  
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: $( - \infty , \infty )$ 
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: $( - \infty , \infty )$

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 $- 8$ .

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

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. - $f ( x ) = 2 x ^ { 4 } - x ^ { 3 } - 15 x ^ { 2 } + 3 x ; k = - 3$

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

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. - $y = ( x + 1 ) ^ { 2 } + 5$

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?

Find the equation that the given graph represents. -

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. - $f ( x ) = x ^ { 2 } + 4$

Review of Basic Concepts

Equations and Inequalities

Graphs and Functions

Inverse, Exponential, and Logarithmic Functions

Trigonometric Functions

The Circular Functions and Their Graphs

Trigonometric Identities and Equations

Applications of Trigonometry

Systems and Matrices

Analytic Geometry

Further Topics in Algebra

Filters

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