Question 1

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

Accepted Answer

A)  
B)   
C)   
D)

Question 2

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

Accepted Answer

A)    2,4,−1;f(x) =(2x−1) (4x−1) (x+1) 2,4 , - 1 ; f ( x )  = ( 2 x - 1 )  ( 4 x - 1 )  ( x + 1 ) 2,4,−1;f(x) =(2x−1) (4x−1) (x+1)  
B)    18,4,−1;f(x) =(8x−1) (x−4) (x+1) \frac { 1 } { 8 } , 4 , - 1 ; f ( x )  = ( 8 x - 1 )  ( x - 4 )  ( x + 1 ) 81​,4,−1;f(x) =(8x−1) (x−4) (x+1)  
C)    18,1,−1;f(x) =(8x−1) (x−1) (x+1) \frac { 1 } { 8 } , 1 , - 1 ; f ( x )  = ( 8 x - 1 )  ( x - 1 )  ( x + 1 ) 81​,1,−1;f(x) =(8x−1) (x−1) (x+1)  
D)    12,14,−1;f(x) =(2x−1) (4x−1) (x+1) \frac { 1 } { 2 } , \frac { 1 } { 4 } , - 1 ; f ( x )  = ( 2 x - 1 )  ( 4 x - 1 )  ( x + 1 ) 21​,41​,−1;f(x) =(2x−1) (4x−1) (x+1)

Question 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

Accepted Answer

A)    −3,1+213i,1−213i- 3,1 + 2 \sqrt { 13 } i , 1 - 2 \sqrt { 13 } i−3,1+213c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​i,1−213c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​i 
B)    −3,3+4i,3−4i- 3,3 + 4 i , 3 - 4 i−3,3+4i,3−4i 
C)    −3,1+2i,1−2i- 3,1 + 2 i , 1 - 2 i−3,1+2i,1−2i 
D)    −3,3+2i,3−2i- 3,3 + 2 i , 3 - 2 i−3,3+2i,3−2i

Question 4

Sketch the graph of the polynomial function.
-

Accepted Answer

A)  
B)  
C)  
D)

Question 5

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) =2x3+8x2−10x+12P ( x )  = 2 x ^ { 3 } + 8 x ^ { 2 } - 10 x + 12P(x) =2x3+8x2−10x+12 
B)    P(x) =−2x3+4x2+10x−12P ( x )  = - 2 x ^ { 3 } + 4 x ^ { 2 } + 10 x - 12P(x) =−2x3+4x2+10x−12 
C)    P(x) =2x3−4x2−10x+12P ( x )  = 2 x ^ { 3 } - 4 x ^ { 2 } - 10 x + 12P(x) =2x3−4x2−10x+12 
D)    P(x) =−2x3−8x2+10x−12P ( x )  = - 2 x ^ { 3 } - 8 x ^ { 2 } + 10 x - 12P(x) =−2x3−8x2+10x−12

Question 6

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

Accepted Answer

A)    f(x) =x2+x−12f ( x )  = x ^ { 2 } + x - 12f(x) =x2+x−12 
B)    f(x) =2x2+2x−24f ( x )  = 2 x ^ { 2 } + 2 x - 24f(x) =2x2+2x−24 
C)    f(x) =x2+5x−24f ( x )  = x ^ { 2 } + 5 x - 24f(x) =x2+5x−24 
D)    f(x) =x2+x−24f ( x )  = x ^ { 2 } + x - 24f(x) =x2+x−24

Question 7

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 8

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

Accepted Answer

A)    −3,4,−5;f(x) =(x+3) (x−4) (x+5) - 3,4 , - 5 ; f ( x )  = ( x + 3 )  ( x - 4 )  ( x + 5 ) −3,4,−5;f(x) =(x+3) (x−4) (x+5)  
B)    −112,1,−5;f(x) =(12x+1) (x−1) (x+5) - \frac { 1 } { 12 } , 1 , - 5 ; f ( x )  = ( 12 x + 1 )  ( x - 1 )  ( x + 5 ) −121​,1,−5;f(x) =(12x+1) (x−1) (x+5)  
C)    −3,4,−5;f(x) =(3x+1) (4x−1) (x+5) - 3,4 , - 5 ; f ( x )  = ( 3 x + 1 )  ( 4 x - 1 )  ( x + 5 ) −3,4,−5;f(x) =(3x+1) (4x−1) (x+5)  
D)    −13,14,−5;f(x) =(3x+1) (4x−1) (x+5) - \frac { 1 } { 3 } , \frac { 1 } { 4 } , - 5 ; f ( x )  = ( 3 x + 1 )  ( 4 x - 1 )  ( x + 5 ) −31​,41​,−5;f(x) =(3x+1) (4x−1) (x+5)

Question 9

Use synthetic division  f(x)  by x−k for the given value of k. Then express f(x)  in the form f(x) =(x−k) q(x) +rf ( x )  	ext { by } x - k 	ext { for the given value of } k 	ext {. Then express } f ( x )  	ext { in the form } f ( x )  = ( x - k )  q ( x )  + rf(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) =−6x3+2x2+5x−10;k=2f ( x )  = - 6 x ^ { 3 } + 2 x ^ { 2 } + 5 x - 10 ; \quad k = 2f(x) =−6x3+2x2+5x−10;k=2

Accepted Answer

A)    f(x) =(x+2) (−6x2+10x+25) −40f ( x )  = ( x + 2 )  \left( - 6 x ^ { 2 } + 10 x + 25 \right)  - 40f(x) =(x+2) (−6x2+10x+25) −40 
B)    f(x) =(x−2) (−6x2−10x+15) −40f ( x )  = ( x - 2 )  \left( - 6 x ^ { 2 } - 10 x + 15 \right)  - 40f(x) =(x−2) (−6x2−10x+15) −40 
C)    f(x) =(x−2) (−6x2+10x+25) −40f ( x )  = ( x - 2 )  \left( - 6 x ^ { 2 } + 10 x + 25 \right)  - 40f(x) =(x−2) (−6x2+10x+25) −40 
D)    f(x) =(x−2) (−6x2−10x−15) −40f ( x )  = ( x - 2 )  \left( - 6 x ^ { 2 } - 10 x - 15 \right)  - 40f(x) =(x−2) (−6x2−10x−15) −40

Question 10

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 11

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

Accepted Answer

A)  
B)  
C)  
D)

Question 12

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.

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.

Question 13

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

Accepted Answer

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

Question 14

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

Accepted Answer

A)    −2- 2−2  (multiplicity 3) , 0 (multiplicity 2) , 6 (multiplicity 1) 
B)    −2- 2−2  (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 )

Question 15

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

Accepted Answer

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

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

Question 17

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​

Accepted Answer

A)  
B)  
C)  
D)

Question 18

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 19

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

Accepted Answer

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

Question 20

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

Accepted Answer

A)  
B)  
C)  
D)

Exam 4: Polynomials and Rational Functions

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

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.

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

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

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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

Correct AnswerverifiedShow Answer

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

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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

Correct AnswerverifiedShow Answer

Sketch the graph of the polynomial function. -

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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

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

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​

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

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?

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
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Correct Answer
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Correct Answer
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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.

Correct Answer
verified

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. - $f ( x ) = 8 x ^ { 5 } - 7 x ^ { 4 } + 8 x ^ { 3 } - 6$

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

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

Correct Answer
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Correct Answer
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Sketch the graph of the polynomial function. - $f ( x ) = - \frac { 1 } { 3 } ( x + 2 ) ^ { 3 }$ $Sketch the graph of the polynomial function. - f ( x ) = - \frac { 1 } { 3 } ( x + 2 ) ^ { 3 } A) B) C) D)$

Correct Answer
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Correct Answer
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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

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

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

Correct Answer
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Sketch the graph of the parabola. - $y = 2 ( x + 3 ) ^ { 2 } + 4$ $Sketch the graph of the parabola. - y = 2 ( x + 3 ) ^ { 2 } + 4 A) B) C) D)$

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

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

Correct Answer
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