Question 1

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

Accepted Answer

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 
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 2

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

Accepted Answer

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) 
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 3

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

Accepted Answer

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) 
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 4

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

Accepted Answer

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

Question 5

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

Accepted Answer

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)  $( x + 5 i ) ( x - 5 ) ( x + \sqrt { 5 i } ) ( x - \sqrt { 5 } )$ 
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 6

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

Accepted Answer

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

Question 7

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

Accepted Answer

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 
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 8

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

Accepted Answer

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 
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 9

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

Accepted Answer

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

Question 10

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

Accepted Answer

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 
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

Question 11

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

Accepted Answer

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

Accepted Answer

A) ±5i, $\frac { 8 } { 7 }$ 
B) ±5i, $- \frac { 8 } { 7 }$ 
C) ±5i,-8 
D) ±5i, $- \frac { 7 } { 8 }$ 
E) ±5i, $\pm \frac { 8 } { 7 } i$ 
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 13

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

Accepted Answer

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 
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 14

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

Accepted Answer

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

Question 15

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

Accepted Answer

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 
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

Question 16

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

Accepted Answer

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 
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 17

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

Accepted Answer

A) x = 0,1,-4 
B) x = 0,-1,4 
C) x = -1,64 
D) x = 1,-64 
E) x = 0,-1,4, $- 2 - 2 \sqrt { 3 } i$ , $- 2 + 2 \sqrt { 3 } i$ 
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 18

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

Accepted Answer

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)  $\left( x ^ { 2 } + 9 \right) ( x + \sqrt { 3 } ) ( x + \sqrt { 3 } )$ 
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 19

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

Accepted Answer

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) 
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 20

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

Accepted Answer

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 
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

Exam 15: Zeros of Polynomial Functions

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Rectangular Coordinates

Graphs of Equations

Linear Equations in Two Variables

Functions

Analyzing Graphs of Functions

A Library of Parent Functions

Transformations of Functions

Combinations of Functions Composite Functions

Inverse Functions

Mathematical Modeling and Variation

Quadratic Functions and Models

Polynomial Functions of Higher Degree

Polynomial and Synthetic Division

Complex Numbers

Rational Functions

Nonlinear Inequalities

Exponential Functions and Their Graphs

Logarithmic Functions and Their Graphs

Properties of Logarithms

Exponential and Logarithmic Equations

Exponential and Logarithmic Models

Radian and Degree Measure

Trigonometric Functions The Unit Circle

Right Triangle Trigonometry

Trigonometric Functions of Any Angle

Graphs of Sine and Cosine Functions

Graphs of Other Trigonometric Functions

Inverse Trigonometric Functions

Applications and Models

Using Fundamental Identities

Verifying Trigonometric Equations

Solving Trigonometric Equations

Sum and Difference Formulas

Multiple Angle and Product to Sum Formulas

Law of Sines

Law of Cosines

Vectors in the Plane

Vectors and Dot Products

Trigonometric Form of a Complex Number

Linear and Nonlinear Systems of Equations

Two Variable Linear Systems

Multivariable Linear Systems

Partial Fractions

Systems of Inequalities

Linear Programming

Matrices and Systems of Equations

Operations With Matrices

The Inverse of a Square Matrix

The Determinant of a Square Matrix

Applications of Matrices and Determinants

Sequences and Series

Arithmetic Sequences and Partial Sums

Geometric Sequences and Series

Mathematical Induction

The Binomial Theorem

Counting Principles

Probability

Lines

Introduction to Conics Parabolas

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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