Deck 15: Zeros of Polynomial Functions

Question 1

Find all real solutions of the polynomial equation x⁴ - 7x³ + 42x - 36 = 0. ?
?

A)x = 1,6, $\pm \sqrt { 6 }$
B)x = 1,36
C)x = 1, $\pm \sqrt { 6 }$
D)x = 1,-7,-6
E)x = 1,-36,12

Answer 1

x = 1,6, $\pm \sqrt { 6 }$

Question 2

A bulk food storage bin with dimensions 3 feet by 4 feet by 5 feet needs to be increased in size to hold five times as much food as the current bin.(Assume each dimension is increased by the same amount. ).Write a function that represents the volume of the new bin. ​

A)​V(x)= x³ + 12x² + 47x + 60 = -300
B)V(x)= x³ + 12x² + 47x + 60 = 300
C)V(x)= x³ + 12x² + 47x = 300
D)V(x)= x³ + 12x² - 47x - 60 = 300
E)V(x)= x³ - 12x² + 47x - 60 = 300

Answer 2

V(x)= x³ + 12x² + 47x + 60 = 300

Question 3

Find a polynomial function with real coefficients that has the given zeros. ​
1,3i
​

A)x³ - x² - 9x - 9
B)x³ + x² + 9x - 9
C)x³ - x² + 9x - 9
D)x³ + x² + 9x + 9
E)x³ - x² + 9x + 9

Answer 3

x³ - x² + 9x - 9

Question 4

Find all the rational zeros of the function. ​
X³ - 18x² + 108x - 216
​

A)-6
B)0,-1
C)0,6
D)6
E)±6

Question 5

Find all real solutions of the polynomial equation 6x⁶ + x⁵ + 29x⁴ + 5x³ + 19x² + 4x - 4 = 0.

A)x = $\frac { 1 } { 3 }$ , $- \frac { 1 } { 2 }$ ,±1
B)x = $- \frac { 1 } { 3 }$ , $\frac { 1 } { 2 }$ ,±2
C)x = $\frac { 1 } { 3 }$ , $- \frac { 1 } { 2 }$
D)x = ± $\frac { 1 } { 3 }$ ,± $\frac { 1 } { 2 }$ ,±1
E)x = ± $\frac { 1 } { 3 }$ ,± $\frac { 1 } { 2 }$ ,±2

Question 6

Use Descartes' Rule of Signs to determine the possible number of positive and negative zeros of f(x)= 6x³ + 3x² + 3x + 4.

A)no positive reals;no negative reals
B)1 positive real;3 negative reals or 1 negative real
C)0 positive real;3 negative reals or 1 negative real
D)3 positive reals or 1 positive real;no negative reals
E)3 positive reals or 1 positive real;3 negative reals or 1 negative real

Question 7

Find all the real zeros of f(x)= 4x³ - 12x² - 15x - 4.

A)x = $\frac { 1 } { 4 }$ ,2
B)x = $- \frac { 1 } { 2 }$ , $- \frac { 1 } { 4 }$
C)x = $\frac { 1 } { 2 }$ , $- \frac { 1 } { 4 }$
D)x = $- \frac { 1 } { 2 }$ ,4
E)x = $- \frac { 1 } { 2 }$ ,-4

Question 8

Find all the rational zeros of the function f(x)= 2x⁴ - 12x³ - 34x² + 300x - 400.

A)x = -4,-5,5, $- \frac { 1 } { 2 }$
B)x = 4,-5,5,2
C)x = 2,20, $- \frac { 5 } { 2 }$ ,2
D)x = 2,20,-5
E)x = $\frac { 4 } { 5 }$ , $- \frac { 5 } { 2 }$ ,2,5

Question 9

A rectangular package to be sent by a delivery service can have a maximum combined length and girth (perimeter of a cross section)of 168 inches.​ ​ Write a function that represents the volume of the package.
​

A)V(x)= - 4x²(42 + x)
B)V(x)= 4x²(42 + x)
C)V(x)= 4x²(42 - x)
D)V(x)= 4x²(-42 - x)
E)V(x)= - 4x²(42 - x)

Question 10

Find all the zeros of the function. ​
(t - 4)(t - 3)(t - 6i)(t + 6i)
​

A)-4,-3,±6i
B)4,3,6i
C)4,3,±6i
D)-4,3,±6i
E)4,-3,±6i

Question 11

Use Descartes' Rule of Signs to determine the possible number of positive and negative zeros of f(x)= x⁵ + 2x.

A)1 positive real;3 negative reals or 1 negative real
B)no positive reals;no negative reals
C)1 positive real;5 negative reals or 3 negative reals or 1 negative real
D)5 positive reals or 3 positive reals or 1 positive real;no negative reals
E)3 positive reals or 1 positive real;3 negative reals or 1 negative real

Question 12

Use Descartes' Rule of Signs to determine the possible number of positive and negative zeros of f(x)= 6x³ - 5x² + 6x - 6.

A)3 positive reals or 1 positive real;3 negative reals or 1 negative real
B)no positive reals;no negative reals
C)3 positive reals or 1 positive real;1 negative real
D)1 positive real;3 negative reals or 1 negative real
E)3 positive reals or 1 positive real;no negative reals

Question 13

Find all the zeros of the function. ​
X(x - 5)²
​

A)0,5
B)1,5
C)±5
D)1,-5
E)0,-5

Question 14

Find all the rational zeros of the function. ​
X³ + 18x² + 105x + 200
​

A)-8,5
B)5,8
C)-8,-5
D)-8,-5,5
E)-5,8

Question 15

Use Descartes's Rule of Signs to determine the possible numbers of positive and negative zeros of the function. ​
4x⁵ - 8x
​

A)One negative zero
B)One negative zero or three positive zeros
C)Two negative zeros or no positive zero
D)One positive zero,one negative zero
E)One negative zero,four negative zeros

Question 16

Find all the rational zeros of the function f(x)= 2x⁵ - 9x⁴ + 9x³ - 3x² + 7x + 6.

A)x = $- \frac { 1 } { 2 }$ ,3,2
B)x = $- \frac { 2 } { 3 }$ ,-1,2
C)x = $\frac { 1 } { 2 }$ ,3,1
D)x = $- \frac { 1 } { 2 }$ , $\frac { 3 } { 2 }$
E)x = $- \frac { 1 } { 2 }$ , $\frac { 3 } { 2 }$ ,2

Question 17

Find all the zeros of the function.
​
(x - 2)(x + 6)³
​
a.-2,6
b.

Question 18

Find all the real zeros of f(x)= 3x³ - 9x² + 3x - 9.

A)x = 3
B)x = 3, $- \frac { 1 } { 3 }$
C)x = $\frac { 1 } { 3 }$ , $- \frac { 1 } { 9 }$
D)x = ±1,3
E)x = 3,1,-9

Question 19

Find all the rational zeros of the function. ​
X³ - 7x² + 14x - 8
​

A)-1,-2,-4
B)1,2,4
C)1,-2,4
D)-1,2,4
E)​-1,2,-4

Question 20

A manufacturer wants to enlarge an existing manufacturing facility such that the total floor area is 1.5 times that of the current facility.The floor area of the current facility is rectangular and measures 290 feet by 80 feet.The manufacturer wants to increase each dimension by the same amount. ​

A)A(x)= (290 - x)(80 - x)= 34,800 ft²
B)A(x)= (290 - x)(80 + x)= 34,800 ft²
C)A(x)= (290 + x)(80 - x)= 34,800 ft²
D)A(x)= (290 + x)(80 + x)= 34,800 ft²
E)A(x)= (290 - x)(80 - x)= -34,800 ft²

Question 21

Find a polynomial function with real coefficients that has the given zeros. ​
6,2 + i
​

A)x³ - 10x² - 29x - 30
B)x³ + 10x² + 29x - 30
C)x³ - 10x² + 29x - 30
D)x³ - 10x² + 29x + 30
E)x³ + 10x² + 29x + 30

Question 22

Find a polynomial function with real coefficients that has the given zeros. ​
2,4 + 3i
​

A)x³ - 10x² + 41x + 50
B)x³ + 10x² + 41x + 50
C)x³ - 10x² + 41x - 50
D)x³ - 10x² - 41x - 50
E)x³ + 10x² + 41x - 50

Question 23

Find all the zeros of the function and write the polynomial as a product of linear factors. ​
X³ - 6x² + x + 164
​
​

A)-4,5 ± 4i; (x + 4)(x - 5 + 4i)(x - 5 - 4i)
B)-4,5 ± 4i; (x + 4)(x - 5 + 4i)(x + 5 - 4i)
C)4,5 ± 4i; (x - 4)(x - 5 + 4i)(x - 5 - 4i)
D)-4,5 ± 4i; (x + 4)(x + 5 + 4i)(x + 5 - 4i)
E)-4,5 ± 4i; (x + 4)(x - 5 - 4i)(x - 5 - 4i)

Question 24

Use the given zero to find all the zeros of the function. ​
Function Zero
X³ - 3x² + 16x - 48 4i
​

A)±4i,±3
B)±4i,-3
C)±4i,3
D)4,±3
E)±4i,±4

Question 25

Write the polynomial as the product of linear and quadratic factors that are irreducible over the reals. ?
F(x)= x⁴ + 42x² - 343
?

A) $\left( x ^ { 2 } + 49 \right) ( x + \sqrt { 7 } ) ( x - 7 )$
B) $\left( x ^ { 2 } + 49 \right) ( x + \sqrt { 7 } ) ( x + \sqrt { 7 } )$
C) $\left( x ^ { 2 } + 49 \right) ( x - \sqrt { 7 } ) ( x - \sqrt { 7 } )$
D) $\left( x ^ { 2 } + 49 \right) ( x + \sqrt { 7 } ) ( x - \sqrt { 7 } )$
E) $\left( x ^ { 2 } + 49 \right) ( x + 7 ) ( x - \sqrt { 7 } )$

Question 26

Write the polynomial in completely factored form. ?
F(x)= x⁴ + 20x² - 125
?

A) $( x + 5 ) ( x - 5 ) ( x + \sqrt { 5 } ) ( x - \sqrt { 5 } )$
B) $$( x + 5 ) ( x - 5 i ) ( x + \sqrt { 5 } ) ( x - \sqrt { 5 } i )$$
C) $$( x + 5 ) ( x - 5 ) ( x + \sqrt { 5 } i ) ( x - \sqrt { 5 } i )$$
D) $( x + 5 i ) ( x - 5 i ) ( x + \sqrt { 5 } ) ( x - \sqrt { 5 } )$
E) $( x + 5 i ) ( x - 5 ) ( x + \sqrt { 5 i } ) ( x - \sqrt { 5 } )$

Question 27

Find all the zeros of the function and write the polynomial as a product of linear factors. ​
Y⁴ - 2401
​

A)±7; (y - 7)(y + 7)
B)±7,±7i; (y - 7)(y + 7)(y - 7i)(y + 7i)
C)7,7i; (y + 7)(y + 7)(y - 7i)(y - 7i)
D)±7i; (y - 7i)(y + 7i)
E)±7,±7i; (y - 7)(y - 7)(y + 7i)(y + 7i)

Question 28

Find a polynomial function with real coefficients that has the given zeros. ​
6,-6i
​

A)x³ - 6x² - 36x - 216
B)x³ - 6x² + 36x + 216
C)x³ + 6x² + 36x + 216
D)x³ - 6x² + 36x - 216
E)x³ + 6x² + 36x - 216

Question 29

Use the given zero to find all the zeros of the function. ?
Function Zero
7x³ + 8x² + 175x + 200 5i
?
?

A)±5i, $\frac { 8 } { 7 }$
B)±5i, $- \frac { 8 } { 7 }$
C)±5i,-8
D)±5i, $- \frac { 7 } { 8 }$
E)±5i, $\pm \frac { 8 } { 7 } i$

Question 30

Find all zeros of the function f(x)= (x - 4)(x + 3)[x + (4 + 3i)][x - (4 - 3i)].

A)x = 4,-3,-4 - 3i,4 - 3i
B)x = -4,3,4 - 3i,4 + 3i
C)x = -4,3,4 + 3i,-4 - 3i
D)x = 4,-3,4 + 3i,-4 + 3i
E)x = 4,-3,-4 - 3i,4 + 3i

Question 31

Find all the zeros of the function and write the polynomial as a product of linear factors. ​
X⁴ - 81
​

A)3,3i; (x + 3)(x + 3)(x - 3i)(x - 3i)
B)±3i; (x - 3i)(x + 3i)
C)±3,±3i; (x - 3)(x - 3)(x + 3i)(x + 3i)
D)±3; (x - 3)(x + 3)
E)±3,±3i; (x - 3)(x + 3)(x - 3i)(x + 3i)

Question 32

Write the number of rational and irrational zeros of the given cubic function. ​
X³ - 2
​

A)Rational zeros: 1;irrational zeros: 0
B)Rational zeros: 1;irrational zeros: 2
C)Rational zeros: 1;irrational zeros: 1
D)Rational zeros: 0;irrational zeros: 1
E)Rational zeros: 0;irrational zeros: 0

Question 33

Write the polynomial as the product of factors that are irreducible over the rationals. ?
F(x)= x⁴ + 6x² - 27
?

A) $\left( x ^ { 2 } + 9 \right) \left( x ^ { 4 } - 3 \right)$
B) $\left( x ^ { 2 } + 9 \right) \left( x ^ { 2 } - 3 \right)$
C) $( x + 9 ) \left( x ^ { 2 } - 3 \right)$
D) $( x + 6 ) \left( x ^ { 2 } - 3 \right)$
E) $\left( x ^ { 2 } + 9 \right) ( x + \sqrt { 3 } ) ( x + \sqrt { 3 } )$

Question 34

Find all zeros of the function f(x)= x²(x + 1)(x³ - 64).

A)x = 0,1,-4
B)x = 0,-1,4
C)x = -1,64
D)x = 1,-64
E)x = 0,-1,4, $- 2 - 2 \sqrt { 3 } i$ , $- 2 + 2 \sqrt { 3 } i$

Question 35

Find all the zeros of the function and write the polynomial as a product of linear factors. ​
X² + 49
​

A)7i; (x + 7i)(x - 7i)
B)-7i; (x + 7i)(x - 7i)
C)±7i; (x + 7)(x - 7)
D)-7i; (x + 7)(x - 7)
E)±7i; (x + 7i)(x - 7i)

Question 36

Find all the zeros of the function. ​
X³ + 7x² + 32x + 60
​

A)-4,-3 ± 2i
B)-3,-2 ± 4i
C)-3,2 ± 4i
D)-3,2 ± i
E)-3,-4 ± 2i

Question 37

Find all zeros of the function f(x)= (x - 4)(x + 4i)(x - 4i).

A)x = 4,-4i,4i
B)x = 4
C)x = 4,4i
D)x = 4,-4,4
E)x = -4,-4i,4i

Question 38

Use the given zero to find all the zeros of the function. ​
Function Zero
X³ + 13x² + 59x + 87 -5 - 2i
​
​

A)3,5 ± 2i
B)±3i,-5 ± 2i
C)-3,5 ± 2i
D)-3,-5 ± 2i
E)3,-5 ± 2i

Question 39

Find all the zeros of the function and write the polynomial as a product of linear factors. ​
X³ - 10x² + 29x - 30
​

A)2 ± i;6; (x - 2 + i)(x + 2 - i)(x - 6)
B)2 ± i;6; (x - 2 + i)(x - 2 + i)(x - 6)
C)-2 ± i;6; (x + 2 + i)(x + 2 - i)(x - 6)
D)2 ± i;-6; (x - 2 + i)(x - 2 - i)(x + 6)
E)2 ± i;6; (x - 2 + i)(x - 2 - i)(x - 6)

Question 40

Find all the zeros of the function and write the polynomial as a product of linear factors. ​
X⁴ + 13x² + 36
​

A)±3i,±2i; (x + 3i)(x - 3i)(x + 2)(x - 2)
B)±3i,±2i; (x + 3i)(x - 3i)(x + 2i)(x - 2i)
C)-3i,-2i; (x - 3i)(x - 3i)(x - 2i)(x - 2i)
D)3i,2i; (x + 3i)(x + 3i)(x + 2i)(x + 2i)
E)±3i,±2i; (x + 3)(x - 3)(x + 2i)(x - 2i)

Question 41

Use the zero or root feature of a graphing utility to approximate the zeros of the function f(x)= x⁶ - 9x⁴ + 11x² + 21 accurate to the nearest thousandth.

A)2.646,1.732
B)±1,-2.646,1.732
C)±i,-2.646,-1.732
D)±1,2.646,1.732
E)±i,±2.646,±1.732

Question 42

Given 3 + i is a root,determine all other roots of f(x)= x⁴ - 12x³ + 59 x² - 138x + 130.

A)x = 3 - i,3 ± 2i
B)x = 3 - i,2 ± 3i
C)x = 3 + i,3 ± 2i​,2 - i
D)x = 3 - i,2 ± i
E)x = 3 - i,3 - 2i​,2 - i

Question 43

Given -6i is a root,determine all other roots of f(x)= x³ + 2x² + 36x + 72. ​

A)x = -2,±6
B)x = -2,6i
C)x = -6,±2i
D)x = ±2,6i
E)x = ±2,6

Question 44

Write f(x)= x³ - 5x² + 16x - 80 as a product of linear factors.

A)f(x)= (x - 5)(x - 4)²
B)f(x)= (x - 5)(x + 4i)(x - 4i)
C)f(x)= (x - 5)²(x - 4i)
D)f(x)= (x - 5)(x + 4)²
E)f(x)= (x + 5)(x - 5)(x + 4)

Question 45

Find a polynomial with real coefficients that has zeros -1,3i,and -3i.

A)x³ + 9x² + x + 9
B)x³ + x² - 9x - 9
C)x³ + x² + 3x + 3
D)x³ + x² + 9x + 9
E)x³ - x² + 9x - 9

Question 46

Write f(x)= x⁴ - 2x³ - x² - 38 x + 130 as a product of linear factors. ​

A)(x - 3 - i)(x - 3 + i)(x - 2 - 3i)(x - 2 + i)
B)(x - 3 + i)(x - 3 - i)(x - 3 + 2i)(x - 3 - 2i)
C)(x - 3 - i)(x - 2 + 3i)(x - 2 - 3i)(x - 2 + i)
D)(x - 3 - i)(x - 3 + i)(x - 2 - i)(x - 2 + i)
E)(x - 3 + i)(x - 3 - i)(x + 2 + 3i)(x + 2 - 3i)

Question 47

Find a polynomial with the given zeros.? $i , \sqrt { 7 } , - i$ ?

A)? $x ^ { 3 } - 7 x$
B) $x ^ { 3 } - \sqrt { 7 } x ^ { 2 } + x - \sqrt { 7 }$
C) $x ^ { 3 } - \sqrt { 7 } x ^ { 2 } + 7 x - \sqrt { 7 }$
D) $x ^ { 3 } - 7 x ^ { 2 } + x - 7$
E)none of these

Question 48

Given -3i is a root,determine all other roots of f(x)= x³ + 2x² + 9x + 18.

A)x = ±2,3i
B)x = -3,±2i
C)x = -2,±3
D)x = -2,3i
E)x = ±2,3

Question 49

Given 3 + i is a root,determine all other roots of f(x)= x⁵ - 8x⁴ + 24x³ - 32x² + 20x.

A)x = 1 - i,3 - i
B)x = 1 - i,3 - i,0
C)x = 1 + i,3 - i
D)x = 1 ± i,3 - i,0
E)x = 1 + i,3 - i,0