Question 1

Find a polynomial $f ( x )$ of degree $3$ that has the indicated zeros and satisfies the given condition. $2,2 i , - 2 i ; f ( 1 ) = - 15$

Accepted Answer

A)  $f ( x ) = 48 x ^ { 3 } - 9 x ^ { 2 } + 12 x - 24$ 
B)  $f ( x ) = 2 x ^ { 3 } - 2 x ^ { 2 } + 2 x - 30$ 
C)  $f ( x ) = 2 x ^ { 3 } + 2 x ^ { 2 } + 2 x - 30$ 
D)  $f ( x ) = 3 x ^ { 3 } - 6 x ^ { 2 } + 12 x - 12$ 
E)  $f ( x ) = 3 x ^ { 3 } - 6 x ^ { 2 } + 12 x - 24$ 
A)  $f ( x ) = 48 x ^ { 3 } - 9 x ^ { 2 } + 12 x - 24$ 
B)  $f ( x ) = 2 x ^ { 3 } - 2 x ^ { 2 } + 2 x - 30$ 
C)  $f ( x ) = 2 x ^ { 3 } + 2 x ^ { 2 } + 2 x - 30$ 
D)  $f ( x ) = 3 x ^ { 3 } - 6 x ^ { 2 } + 12 x - 12$ 
E)  $f ( x ) = 3 x ^ { 3 } - 6 x ^ { 2 } + 12 x - 24$

Question 2

Find the domain $D$ of $f$ . $$f ( x ) = \frac { 4 } { x }$$

Accepted Answer

A)  $( - \infty , 4 ) \cup ( 4 , \infty )$ 
B)  $( - \infty , \infty )$ 
C)  $( - \infty , - 4 ) \cup ( 4 , \infty )$ 
D)  $( - \infty , 0 ) \cup ( 0 , \infty )$ 
E)  $( - \infty , 0 ) \cup ( 4 , \infty )$ 
A)  $( - \infty , 4 ) \cup ( 4 , \infty )$ 
B)  $( - \infty , \infty )$ 
C)  $( - \infty , - 4 ) \cup ( 4 , \infty )$ 
D)  $( - \infty , 0 ) \cup ( 0 , \infty )$ 
E)  $( - \infty , 0 ) \cup ( 4 , \infty )$

Question 3

Applying the first theorem on bounds for real zeros of polynomials, determine the smallest and largest integers that are upper and lower bounds, respectively, for the real solutions of the equation. With the aid of a graphing utility, discuss the validity of the bounds. $$3 x ^ { 4 } - 7 x ^ { 3 } - 5 x - 6 = 0$$

Accepted Answer

A)  The upper bound is $$5$$ , the lower bound is $- 3$ . 
B)  The upper bound is $3$ , the lower bound is $- 1$ . 
C)  The upper bound is $0$ , the lower bound is $- 2$ . 
D)  The upper bound is $1$ , the lower bound is $$- 3$$ . 
E)  The upper bound is $2$ , the lower bound is $- 4$ . 
A)  The upper bound is $$5$$ , the lower bound is $- 3$ . 
B)  The upper bound is $3$ , the lower bound is $- 1$ . 
C)  The upper bound is $0$ , the lower bound is $- 2$ . 
D)  The upper bound is $1$ , the lower bound is $$- 3$$ . 
E)  The upper bound is $2$ , the lower bound is $- 4$ .

Question 4

Use synthetic division to decide whether $c$ is a zero of $f ( x )$ . $f ( x ) = 6 x ^ { 3 } - 3 x ^ { 2 } + 12 x - 6$ ; $c = \frac { 1 } { 2 }$

Accepted Answer

A)  x - c is a zero 
B)  x - c is not a zero 
A)  x - c is a zero 
B)  x - c is not a zero

Find a polynomial $f ( x )$ of degree $3$ that has the indicated zeros and satisfies the given condition. $2,2 i , - 2 i ; f ( 1 ) = - 15$

Find the domain $D$ of $f$ . $f ( x ) = \frac { 4 } { x }$

Use synthetic division to decide whether $c$ is a zero of $f ( x )$ . $f ( x ) = 6 x ^ { 3 } - 3 x ^ { 2 } + 12 x - 6$ ; $c = \frac { 1 } { 2 }$

Topics From Algebra

Functions and Graphs

Inverse, Exponential, and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometry

Systems of Equations and Inequalities

Sequences, Series, and Probability

Topics From Analytical Geometry

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Exam 3: Polynomial and Rational Functions

Find a polynomial f(x)f ( x )f(x) of degree 333 that has the indicated zeros and satisfies the given condition. 2,2i,−2i;f(1)=−152,2 i , - 2 i ; f ( 1 ) = - 152,2i,−2i;f(1)=−15

Find the domain DDD of fff . f(x)=4xf ( x ) = \frac { 4 } { x }f(x)=x4​

Use synthetic division to decide whether ccc is a zero of f(x)f ( x )f(x) . f(x)=6x3−3x2+12x−6f ( x ) = 6 x ^ { 3 } - 3 x ^ { 2 } + 12 x - 6f(x)=6x3−3x2+12x−6 ; c=12c = \frac { 1 } { 2 }c=21​

Topics From Algebra

Functions and Graphs

Inverse, Exponential, and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometry

Systems of Equations and Inequalities

Sequences, Series, and Probability

Topics From Analytical Geometry

Filters

Find a polynomial $f ( x )$ of degree $3$ that has the indicated zeros and satisfies the given condition. $2,2 i , - 2 i ; f ( 1 ) = - 15$

Find the domain $D$ of $f$ . $f ( x ) = \frac { 4 } { x }$

Use synthetic division to decide whether $c$ is a zero of $f ( x )$ . $f ( x ) = 6 x ^ { 3 } - 3 x ^ { 2 } + 12 x - 6$ ; $c = \frac { 1 } { 2 }$