Question 1

Sketch the graph of the polynomial function.
-

Accepted Answer

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

Question 2

Find a polynomial of degree 3 with real coefficients that satisfies the given conditions.
Zeros of  1,-2,3  and  P(2)=8

Accepted Answer

A)  $P ( x ) = 2 x ^ { 3 } + 8 x ^ { 2 } - 10 x + 12$ 
B)  $P ( x ) = - 2 x ^ { 3 } + 4 x ^ { 2 } + 10 x - 12$ 
C)  $P ( x ) = 2 x ^ { 3 } - 4 x ^ { 2 } - 10 x + 12$ 
D)  $P ( x ) = - 2 x ^ { 3 } - 8 x ^ { 2 } + 10 x - 12$ 
A)  $P ( x ) = 2 x ^ { 3 } + 8 x ^ { 2 } - 10 x + 12$ 
B)  $P ( x ) = - 2 x ^ { 3 } + 4 x ^ { 2 } + 10 x - 12$ 
C)  $P ( x ) = 2 x ^ { 3 } - 4 x ^ { 2 } - 10 x + 12$ 
D)  $P ( x ) = - 2 x ^ { 3 } - 8 x ^ { 2 } + 10 x - 12$

Question 3

Use the intermediate value theorem for polynomials to show that the polynomial function has a real zero between the numbers given.
-$f ( x ) = - 8 x ^ { 4 } - 5 x ^ { 3 } - 9 x - 5 ; - 1$ and 0

Accepted Answer

A)  $f ( - 1 ) = 2$ and $f ( 0 ) = - 4$ 
B)  $f ( - 1 ) = - 1$ and $f ( 0 ) = - 5$ 
C)  $f ( - 1 ) = - 1$ and $f ( 0 ) = 5$ 
D)  $f ( - 1 ) = 1$ and $f ( 0 ) = - 5$ 
A)  $f ( - 1 ) = 2$ and $f ( 0 ) = - 4$ 
B)  $f ( - 1 ) = - 1$ and $f ( 0 ) = - 5$ 
C)  $f ( - 1 ) = - 1$ and $f ( 0 ) = 5$ 
D)  $f ( - 1 ) = 1$ and $f ( 0 ) = - 5$

Question 4

Determine whether the statement is true or false.
-$$\text { If } \mathrm { c } \text { is a complex number, then } 5 \bar { c } = \overline { 5 c }$$

Accepted Answer

A) True 
 B)False

Question 5

Find the equation of the axis of symmetry of the parabola.
-$$f ( x ) = ( x - 5 ) ^ { 2 } - 5$$

Accepted Answer

A)  $y = 0$ 
B)  $x = - 5$ 
C)  $y = - 5$ 
D)  $x = 5$ 
A)  $y = 0$ 
B)  $x = - 5$ 
C)  $y = - 5$ 
D)  $x = 5$

Question 6

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

Accepted Answer

A)  Yes, the boundedness theorem shows that the polynomial has no real zero greater than 3. 
B)  No, the boundedness theorem does not show that the polynomial has no real zero greater than 3. 
A)  Yes, the boundedness theorem shows that the polynomial has no real zero greater than 3. 
B)  No, the boundedness theorem does not show that the polynomial has no real zero greater than 3.

Question 7

Find all complex zeros of the polynomial function. Give exact values. List multiple zeros as necessary.
-$$f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 5 x + 39$$

Accepted Answer

A)  $- 3,1 + 2 \sqrt { 13 } i , 1 - 2 \sqrt { 13 } i$ 
B)  $- 3,3 + 4 i , 3 - 4 i$ 
C)  $- 3,1 + 2 i , 1 - 2 i$ 
D)  $- 3,3 + 2 i , 3 - 2 i$ 
A)  $- 3,1 + 2 \sqrt { 13 } i , 1 - 2 \sqrt { 13 } i$ 
B)  $- 3,3 + 4 i , 3 - 4 i$ 
C)  $- 3,1 + 2 i , 1 - 2 i$ 
D)  $- 3,3 + 2 i , 3 - 2 i$

Question 8

Determine whether the statement is true or false.
-$$\text { If } f ( x ) \text { is a polynomial and } f ( - 5 ) = 0 \text {, then } x + 5 \text { is a factor of } f ( x ) \text {. }$$

Accepted Answer

A) True 
 B)False

Question 9

Use synthetic division $$f ( x ) \text { by } x - k \text { for the given value of } k \text {. Then express } f ( x ) \text { in the form } f ( x ) = ( x - k ) q ( x ) + r$$ for the given value of k.
-$$f ( x ) = - 6 x ^ { 3 } + 2 x ^ { 2 } + 5 x - 10 ; \quad k = 2$$

Accepted Answer

A)  $f ( x ) = ( x + 2 ) \left( - 6 x ^ { 2 } + 10 x + 25 \right) - 40$ 
B)  $f ( x ) = ( x - 2 ) \left( - 6 x ^ { 2 } - 10 x + 15 \right) - 40$ 
C)  $f ( x ) = ( x - 2 ) \left( - 6 x ^ { 2 } + 10 x + 25 \right) - 40$ 
D)  $f ( x ) = ( x - 2 ) \left( - 6 x ^ { 2 } - 10 x - 15 \right) - 40$ 
A)  $f ( x ) = ( x + 2 ) \left( - 6 x ^ { 2 } + 10 x + 25 \right) - 40$ 
B)  $f ( x ) = ( x - 2 ) \left( - 6 x ^ { 2 } - 10 x + 15 \right) - 40$ 
C)  $f ( x ) = ( x - 2 ) \left( - 6 x ^ { 2 } + 10 x + 25 \right) - 40$ 
D)  $f ( x ) = ( x - 2 ) \left( - 6 x ^ { 2 } - 10 x - 15 \right) - 40$

Question 10

Sketch the graph of the polynomial function.
-$$f ( x ) = - \frac { 1 } { 3 } ( x + 2 ) ^ { 3 }$$

Accepted Answer

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

Question 11

Find all rational zeros and factor f(x).
$f ( x ) = 8 x ^ { 3 } + 2 x ^ { 2 } - 5 x + 1$

Accepted Answer

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

Question 12

Sketch the graph of the parabola.
-$$y = 2 ( x + 3 ) ^ { 2 } + 4$$

Accepted Answer

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

Question 13

Find all rational zeros and factor f(x).
$f ( x ) = 12 x ^ { 3 } + 61 x ^ { 2 } + 4 x - 5$

Accepted Answer

A)  $- 3,4 , - 5 ; f ( x ) = ( x + 3 ) ( x - 4 ) ( x + 5 )$ 
B)  $- \frac { 1 } { 12 } , 1 , - 5 ; f ( x ) = ( 12 x + 1 ) ( x - 1 ) ( x + 5 )$ 
C)  $- 3,4 , - 5 ; f ( x ) = ( 3 x + 1 ) ( 4 x - 1 ) ( x + 5 )$ 
D)  $- \frac { 1 } { 3 } , \frac { 1 } { 4 } , - 5 ; f ( x ) = ( 3 x + 1 ) ( 4 x - 1 ) ( x + 5 )$ 
A)  $- 3,4 , - 5 ; f ( x ) = ( x + 3 ) ( x - 4 ) ( x + 5 )$ 
B)  $- \frac { 1 } { 12 } , 1 , - 5 ; f ( x ) = ( 12 x + 1 ) ( x - 1 ) ( x + 5 )$ 
C)  $- 3,4 , - 5 ; f ( x ) = ( 3 x + 1 ) ( 4 x - 1 ) ( x + 5 )$ 
D)  $- \frac { 1 } { 3 } , \frac { 1 } { 4 } , - 5 ; f ( x ) = ( 3 x + 1 ) ( 4 x - 1 ) ( x + 5 )$

Question 14

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function.
-$$f ( x ) = 8 x ^ { 5 } - 7 x ^ { 4 } + 8 x ^ { 3 } - 6$$

Accepted Answer

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

Question 15

Sketch the graph of the rational function.
-$$f ( x ) = \frac { x ^ { 2 } + 3 x - 4 } { 2 x ^ { 2 } + 1 }$$

Accepted Answer

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

Question 16

The weight that a horizontal beam can support varies inversely as the length of the beam. Suppose that a 5-m beam can support 480 kg. How many kilograms can a 2-m beam support?

Accepted Answer

A)  0.0208 kg 
B)  0.0008 kg 
C)  1200 kg 
D)  48 kg 
A)  0.0208 kg 
B)  0.0008 kg 
C)  1200 kg 
D)  48 kg

Question 17

Determine whether the statement is true or false.
-$$\text { If } f ( x ) \text { is a polynomial having only real coefficients and } 3 - 5 i \text { is a zero of } f ( x ) \text {, then } x - 3 - 5 i \text { is a factor of } f ( x ) \text {. }$$

Accepted Answer

A) True 
 B)False

Question 18

Find a quadratic function f having x-intercepts 3 and  -4  and  y-intercept  -24 .

Accepted Answer

A)  $f ( x ) = x ^ { 2 } + x - 12$ 
B)  $f ( x ) = 2 x ^ { 2 } + 2 x - 24$ 
C)  $f ( x ) = x ^ { 2 } + 5 x - 24$ 
D)  $f ( x ) = x ^ { 2 } + x - 24$ 
A)  $f ( x ) = x ^ { 2 } + x - 12$ 
B)  $f ( x ) = 2 x ^ { 2 } + 2 x - 24$ 
C)  $f ( x ) = x ^ { 2 } + 5 x - 24$ 
D)  $f ( x ) = x ^ { 2 } + x - 24$

Question 19

Find all rational zeros and factor f(x).
-$$f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12$$

Accepted Answer

A)  $5,4 , - 1 ; f ( x ) = ( x - 5 ) ( x - 4 ) ( x + 1 )$ 
B)  $4,3 , - 1 ; f ( x ) = ( x - 4 ) ( x - 3 ) ( x + 1 )$ 
C)  $- 5 , - 4,1 ; f ( x ) = ( x + 5 ) ( x + 4 ) ( x - 1 )$ 
D)  $- 4 , - 3,1 ; f ( x ) = ( x + 4 ) ( x + 3 ) ( x - 1 )$ 
A)  $5,4 , - 1 ; f ( x ) = ( x - 5 ) ( x - 4 ) ( x + 1 )$ 
B)  $4,3 , - 1 ; f ( x ) = ( x - 4 ) ( x - 3 ) ( x + 1 )$ 
C)  $- 5 , - 4,1 ; f ( x ) = ( x + 5 ) ( x + 4 ) ( x - 1 )$ 
D)  $- 4 , - 3,1 ; f ( x ) = ( x + 4 ) ( x + 3 ) ( x - 1 )$

Question 20

Find the zeros of the polynomial function and state the multiplicity of each.
-$$f ( x ) = - 5 x ^ { 2 } ( x - 6 ) ( x + 2 ) ^ { 3 }$$

Accepted Answer

A)  $- 2$ (multiplicity 3), 0 (multiplicity 2), 6 (multiplicity 1) 
B)  $- 2$ (multiplicity 3), 6 (multiplicity 1) 
C)  -2 (multiplicity 1), 2 (multiplicity 1), 6 (multiplicity 1 ) 
D)  -2 (multiplicity 3), 0 (multiplicity 2), 2 (multiplicity 1 ), 6 (multiplicity 1 ) 
A)  $- 2$ (multiplicity 3), 0 (multiplicity 2), 6 (multiplicity 1) 
B)  $- 2$ (multiplicity 3), 6 (multiplicity 1) 
C)  -2 (multiplicity 1), 2 (multiplicity 1), 6 (multiplicity 1 ) 
D)  -2 (multiplicity 3), 0 (multiplicity 2), 2 (multiplicity 1 ), 6 (multiplicity 1 )

Sketch the graph of the polynomial function. -

Find a polynomial of degree 3 with real coefficients that satisfies the given conditions. Zeros of 1,-2,3 and P(2)=8

Use the intermediate value theorem for polynomials to show that the polynomial function has a real zero between the numbers given. - $f ( x ) = - 8 x ^ { 4 } - 5 x ^ { 3 } - 9 x - 5 ; - 1$ and 0

Determine whether the statement is true or false. - $\text { If } \mathrm { c } \text { is a complex number, then } 5 \bar { c } = \overline { 5 c }$

Find the equation of the axis of symmetry of the parabola. - $f ( x ) = ( x - 5 ) ^ { 2 } - 5$

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

Find all complex zeros of the polynomial function. Give exact values. List multiple zeros as necessary. - $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 5 x + 39$

Determine whether the statement is true or false. - $\text { If } f ( x ) \text { is a polynomial and } f ( - 5 ) = 0 \text {, then } x + 5 \text { is a factor of } f ( x ) \text {. }$

Use synthetic division $f ( x ) \text { by } x - k \text { for the given value of } k \text {. Then express } f ( x ) \text { in the form } f ( x ) = ( x - k ) q ( x ) + r$ for the given value of k. - $f ( x ) = - 6 x ^ { 3 } + 2 x ^ { 2 } + 5 x - 10 ; \quad k = 2$

Sketch the graph of the polynomial function. - $f ( x ) = - \frac { 1 } { 3 } ( x + 2 ) ^ { 3 }$ $Sketch the graph of the polynomial function. - f ( x ) = - \frac { 1 } { 3 } ( x + 2 ) ^ { 3 }$

Find all rational zeros and factor f(x). $f ( x ) = 8 x ^ { 3 } + 2 x ^ { 2 } - 5 x + 1$

Sketch the graph of the parabola. - $y = 2 ( x + 3 ) ^ { 2 } + 4$ $Sketch the graph of the parabola. - y = 2 ( x + 3 ) ^ { 2 } + 4$

Find all rational zeros and factor f(x). $f ( x ) = 12 x ^ { 3 } + 61 x ^ { 2 } + 4 x - 5$

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function. - $f ( x ) = 8 x ^ { 5 } - 7 x ^ { 4 } + 8 x ^ { 3 } - 6$

Sketch the graph of the rational function. - $f ( x ) = \frac { x ^ { 2 } + 3 x - 4 } { 2 x ^ { 2 } + 1 }$ $Sketch the graph of the rational function. - f ( x ) = \frac { x ^ { 2 } + 3 x - 4 } { 2 x ^ { 2 } + 1 }$

The weight that a horizontal beam can support varies inversely as the length of the beam. Suppose that a 5-m beam can support 480 kg. How many kilograms can a 2-m beam support?

Determine whether the statement is true or false. - $\text { If } f ( x ) \text { is a polynomial having only real coefficients and } 3 - 5 i \text { is a zero of } f ( x ) \text {, then } x - 3 - 5 i \text { is a factor of } f ( x ) \text {. }$

Find a quadratic function f having x-intercepts 3 and -4 and y-intercept -24 .

Find all rational zeros and factor f(x). - $f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12$

Find the zeros of the polynomial function and state the multiplicity of each. - $f ( x ) = - 5 x ^ { 2 } ( x - 6 ) ( x + 2 ) ^ { 3 }$

Review of Basic Concepts

Equations and Inequalities

Graphs and Functions

Inverse, Exponential, and Logarithmic Functions

Systems and Matrices

Arithmetic Sequence: Common Difference and First n Terms

Filters

Exam 4: Polynomials and Rational Functions

Sketch the graph of the polynomial function. -

Find a polynomial of degree 3 with real coefficients that satisfies the given conditions. Zeros of 1,-2,3 and P(2)=8

Use the intermediate value theorem for polynomials to show that the polynomial function has a real zero between the numbers given. - f(x)=−8x4−5x3−9x−5;−1f ( x ) = - 8 x ^ { 4 } - 5 x ^ { 3 } - 9 x - 5 ; - 1f(x)=−8x4−5x3−9x−5;−1 and 0

Determine whether the statement is true or false. - If c is a complex number, then 5cˉ=5c‾\text { If } \mathrm { c } \text { is a complex number, then } 5 \bar { c } = \overline { 5 c } If c is a complex number, then 5cˉ=5c

Find the equation of the axis of symmetry of the parabola. - f(x)=(x−5)2−5f ( x ) = ( x - 5 ) ^ { 2 } - 5f(x)=(x−5)2−5

Use the boundedness theorem to determine whether the polynomial function satisfies the given condition. -The polynomial f(x) 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 24x has no real zero greater than 3.

Find all complex zeros of the polynomial function. Give exact values. List multiple zeros as necessary. - f(x)=x3−3x2−5x+39f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 5 x + 39f(x)=x3−3x2−5x+39

Sketch the graph of the polynomial function. - f(x)=−13(x+2)3f ( x ) = - \frac { 1 } { 3 } ( x + 2 ) ^ { 3 }f(x)=−31​(x+2)3

Find all rational zeros and factor f(x). f(x)=8x3+2x2−5x+1f ( x ) = 8 x ^ { 3 } + 2 x ^ { 2 } - 5 x + 1f(x)=8x3+2x2−5x+1

Sketch the graph of the parabola. - y=2(x+3)2+4y = 2 ( x + 3 ) ^ { 2 } + 4y=2(x+3)2+4

Find all rational zeros and factor f(x). f(x)=12x3+61x2+4x−5f ( x ) = 12 x ^ { 3 } + 61 x ^ { 2 } + 4 x - 5f(x)=12x3+61x2+4x−5

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function. - f(x)=8x5−7x4+8x3−6f ( x ) = 8 x ^ { 5 } - 7 x ^ { 4 } + 8 x ^ { 3 } - 6f(x)=8x5−7x4+8x3−6

Sketch the graph of the rational function. - f(x)=x2+3x−42x2+1f ( x ) = \frac { x ^ { 2 } + 3 x - 4 } { 2 x ^ { 2 } + 1 }f(x)=2x2+1x2+3x−4​

The weight that a horizontal beam can support varies inversely as the length of the beam. Suppose that a 5-m beam can support 480 kg. How many kilograms can a 2-m beam support?

Find a quadratic function f having x-intercepts 3 and -4 and y-intercept -24 .

Find all rational zeros and factor f(x). - f(x)=x3−6x2+5x+12f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12f(x)=x3−6x2+5x+12

Find the zeros of the polynomial function and state the multiplicity of each. - f(x)=−5x2(x−6)(x+2)3f ( x ) = - 5 x ^ { 2 } ( x - 6 ) ( x + 2 ) ^ { 3 }f(x)=−5x2(x−6)(x+2)3

Review of Basic Concepts

Equations and Inequalities

Graphs and Functions

Inverse, Exponential, and Logarithmic Functions

Systems and Matrices

Arithmetic Sequence: Common Difference and First n Terms

Filters

Use the intermediate value theorem for polynomials to show that the polynomial function has a real zero between the numbers given. - $f ( x ) = - 8 x ^ { 4 } - 5 x ^ { 3 } - 9 x - 5 ; - 1$ and 0

Determine whether the statement is true or false. - $\text { If } \mathrm { c } \text { is a complex number, then } 5 \bar { c } = \overline { 5 c }$

Find the equation of the axis of symmetry of the parabola. - $f ( x ) = ( x - 5 ) ^ { 2 } - 5$

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

Find all complex zeros of the polynomial function. Give exact values. List multiple zeros as necessary. - $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 5 x + 39$

Sketch the graph of the polynomial function. - $f ( x ) = - \frac { 1 } { 3 } ( x + 2 ) ^ { 3 }$ $Sketch the graph of the polynomial function. - f ( x ) = - \frac { 1 } { 3 } ( x + 2 ) ^ { 3 }$

Find all rational zeros and factor f(x). $f ( x ) = 8 x ^ { 3 } + 2 x ^ { 2 } - 5 x + 1$

Sketch the graph of the parabola. - $y = 2 ( x + 3 ) ^ { 2 } + 4$ $Sketch the graph of the parabola. - y = 2 ( x + 3 ) ^ { 2 } + 4$

Find all rational zeros and factor f(x). $f ( x ) = 12 x ^ { 3 } + 61 x ^ { 2 } + 4 x - 5$

Use Descartes' Rule of Signs to determine the possible number of positive real zeros and the possible number of negative real zeros for the function. - $f ( x ) = 8 x ^ { 5 } - 7 x ^ { 4 } + 8 x ^ { 3 } - 6$

Sketch the graph of the rational function. - $f ( x ) = \frac { x ^ { 2 } + 3 x - 4 } { 2 x ^ { 2 } + 1 }$ $Sketch the graph of the rational function. - f ( x ) = \frac { x ^ { 2 } + 3 x - 4 } { 2 x ^ { 2 } + 1 }$

Find all rational zeros and factor f(x). - $f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12$

Find the zeros of the polynomial function and state the multiplicity of each. - $f ( x ) = - 5 x ^ { 2 } ( x - 6 ) ( x + 2 ) ^ { 3 }$