Exam 3: Polynomial and Rational Functions

Use Descartes' Rule of Signs and the Rational Zeros Theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers. - $f ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 9 x - 18$

(Multiple Choice)

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Question 21

Use the graph to find the horizontal asymptote, if any, of the function. -

(Multiple Choice)

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Question 22

Find the x- and y-intercepts of f. - $f ( x ) = x ^ { 2 } ( x - 4 ) ( x - 1 )$

(Multiple Choice)

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Question 23

Solve the problem. - $x ( x + 3 ) ( 5 - x ) \geq 0$

(Multiple Choice)

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Question 24

Solve the problem. -Which of the following functions could have this graph?

(Multiple Choice)

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Question 25

Find the domain of the rational function. - $f ( x ) = \frac { - 2 x ( x + 2 ) } { 2 x ^ { 2 } - 3 x - 5 }$

(Multiple Choice)

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Question 26

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not. - $f ( x ) = 5 - \frac { 3 } { x ^ { 6 } }$

(Multiple Choice)

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Question 27

Find the indicated intercept(s) of the graph of the function.2133:2139 - $y$ -intercept of $f ( x ) = \frac { x ^ { 2 } - 5 x } { x ^ { 2 } + 10 x - 13 }$

(Multiple Choice)

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Question 28

Find the x- and y-intercepts of f. - $f ( x ) = ( x + 1 ) ( x - 5 ) ( x - 1 ) ^ { 2 }$

(Multiple Choice)

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Question 29

Solve the problem. - $x ^ { 3 } + 2 x ^ { 2 } - 63 x > 0$

(Multiple Choice)

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Question 30

Solve the problem. -Which of the following polynomial functions might have the graph shown in the illustration below?

(Multiple Choice)

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Question 31

Solve the problem. - $x ^ { 4 } - 12 x ^ { 2 } - 64 > 0$

(Multiple Choice)

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Question 32

Find a bound on the real zeros of the polynomial function. - $f ( x ) = x ^ { 4 } + 2 x ^ { 2 } - 3$

(Multiple Choice)

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Question 33

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept. - $f ( x ) = 2 \left( x ^ { 2 } + 1 \right) \left( x ^ { 2 } + 6 \right) ^ { 2 }$

(Multiple Choice)

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Question 34

Use the Factor Theorem to determine whether x - c is a factor of f. If it is, write f in factored form, that is, write f in the form f(x) = (x - c)(quotient). - $f ( x ) = x ^ { 4 } + 10 x ^ { 3 } + 7 x ^ { 2 } + 61 x - 90 ; c = - 10$

(Multiple Choice)

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Question 35

Solve the problem. - $x ^ { 2 } - 5 x - 6 \leq 0$

(Multiple Choice)

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Question 36

List the potential rational zeros of the polynomial function. Do not find the zeros. - $f ( x ) = - 4 x ^ { 4 } + 3 x ^ { 2 } - 4 x + 6$

(Multiple Choice)

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Question 37

Find the domain of the rational function. - $g ( x ) = \frac { x + 7 } { x ^ { 2 } + 1 }$

(Multiple Choice)

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Question 38

Find the indicated intercept(s) of the graph of the function.2133:2139 - $f ( x ) = x ^ { 4 } - 12 x ^ { 2 } - 64 ; x - 4$

(Multiple Choice)

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Question 39

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not. - $7 ( x - 1 ) ^ { 11 } ( x + 1 ) ^ { 3 }$

(Multiple Choice)

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Question 40

Use Descartes' Rule of Signs and the Rational Zeros Theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers. - $f ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 9 x - 18$

Use the graph to find the horizontal asymptote, if any, of the function. -

Find the x- and y-intercepts of f. - $f ( x ) = x ^ { 2 } ( x - 4 ) ( x - 1 )$

Solve the problem. - $x ( x + 3 ) ( 5 - x ) \geq 0$

Solve the problem. -Which of the following functions could have this graph?

Find the domain of the rational function. - $f ( x ) = \frac { - 2 x ( x + 2 ) } { 2 x ^ { 2 } - 3 x - 5 }$

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not. - $f ( x ) = 5 - \frac { 3 } { x ^ { 6 } }$

Find the indicated intercept(s) of the graph of the function.2133:2139 - $y$ -intercept of $f ( x ) = \frac { x ^ { 2 } - 5 x } { x ^ { 2 } + 10 x - 13 }$

Find the x- and y-intercepts of f. - $f ( x ) = ( x + 1 ) ( x - 5 ) ( x - 1 ) ^ { 2 }$

Solve the problem. - $x ^ { 3 } + 2 x ^ { 2 } - 63 x > 0$

Solve the problem. -Which of the following polynomial functions might have the graph shown in the illustration below?

Solve the problem. - $x ^ { 4 } - 12 x ^ { 2 } - 64 > 0$

Find a bound on the real zeros of the polynomial function. - $f ( x ) = x ^ { 4 } + 2 x ^ { 2 } - 3$

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept. - $f ( x ) = 2 \left( x ^ { 2 } + 1 \right) \left( x ^ { 2 } + 6 \right) ^ { 2 }$

Use the Factor Theorem to determine whether x - c is a factor of f. If it is, write f in factored form, that is, write f in the form f(x) = (x - c)(quotient). - $f ( x ) = x ^ { 4 } + 10 x ^ { 3 } + 7 x ^ { 2 } + 61 x - 90 ; c = - 10$

Solve the problem. - $x ^ { 2 } - 5 x - 6 \leq 0$

List the potential rational zeros of the polynomial function. Do not find the zeros. - $f ( x ) = - 4 x ^ { 4 } + 3 x ^ { 2 } - 4 x + 6$

Find the domain of the rational function. - $g ( x ) = \frac { x + 7 } { x ^ { 2 } + 1 }$

Find the indicated intercept(s) of the graph of the function.2133:2139 - $f ( x ) = x ^ { 4 } - 12 x ^ { 2 } - 64 ; x - 4$

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not. - $7 ( x - 1 ) ^ { 11 } ( x + 1 ) ^ { 3 }$

Functions and Their Graphs

Linear and Quadratic Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometric Functions

Polar Coordinates; Vectors

Analytic Geometry

Systems of Equations and Inequalities

Sequences; Induction; the Binomial Theorem

Counting and Probability

A Preview of Calculus: the Limit, Derivative, and Integral of a Function

Review

Filters

Exam 3: Polynomial and Rational Functions

Use Descartes' Rule of Signs and the Rational Zeros Theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers. - f(x)=x3+2x2−9x−18f ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 9 x - 18f(x)=x3+2x2−9x−18

Use the graph to find the horizontal asymptote, if any, of the function. -

Find the x- and y-intercepts of f. - f(x)=x2(x−4)(x−1)f ( x ) = x ^ { 2 } ( x - 4 ) ( x - 1 )f(x)=x2(x−4)(x−1)

Solve the problem. - x(x+3)(5−x)≥0x ( x + 3 ) ( 5 - x ) \geq 0x(x+3)(5−x)≥0

Solve the problem. -Which of the following functions could have this graph?

Find the domain of the rational function. - f(x)=−2x(x+2)2x2−3x−5f ( x ) = \frac { - 2 x ( x + 2 ) } { 2 x ^ { 2 } - 3 x - 5 }f(x)=2x2−3x−5−2x(x+2)​

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not. - f(x)=5−3x6f ( x ) = 5 - \frac { 3 } { x ^ { 6 } }f(x)=5−x63​

Find the indicated intercept(s) of the graph of the function.2133:2139 - yyy -intercept of f(x)=x2−5xx2+10x−13f ( x ) = \frac { x ^ { 2 } - 5 x } { x ^ { 2 } + 10 x - 13 }f(x)=x2+10x−13x2−5x​

Find the x- and y-intercepts of f. - f(x)=(x+1)(x−5)(x−1)2f ( x ) = ( x + 1 ) ( x - 5 ) ( x - 1 ) ^ { 2 }f(x)=(x+1)(x−5)(x−1)2

Solve the problem. - x3+2x2−63x>0x ^ { 3 } + 2 x ^ { 2 } - 63 x > 0x3+2x2−63x>0

Solve the problem. -Which of the following polynomial functions might have the graph shown in the illustration below?

Solve the problem. - x4−12x2−64>0x ^ { 4 } - 12 x ^ { 2 } - 64 > 0x4−12x2−64>0

Find a bound on the real zeros of the polynomial function. - f(x)=x4+2x2−3f ( x ) = x ^ { 4 } + 2 x ^ { 2 } - 3f(x)=x4+2x2−3

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept. - f(x)=2(x2+1)(x2+6)2f ( x ) = 2 \left( x ^ { 2 } + 1 \right) \left( x ^ { 2 } + 6 \right) ^ { 2 }f(x)=2(x2+1)(x2+6)2

Use the Factor Theorem to determine whether x - c is a factor of f. If it is, write f in factored form, that is, write f in the form f(x) = (x - c)(quotient). - f(x)=x4+10x3+7x2+61x−90;c=−10f ( x ) = x ^ { 4 } + 10 x ^ { 3 } + 7 x ^ { 2 } + 61 x - 90 ; c = - 10f(x)=x4+10x3+7x2+61x−90;c=−10

Solve the problem. - x2−5x−6≤0x ^ { 2 } - 5 x - 6 \leq 0x2−5x−6≤0

List the potential rational zeros of the polynomial function. Do not find the zeros. - f(x)=−4x4+3x2−4x+6f ( x ) = - 4 x ^ { 4 } + 3 x ^ { 2 } - 4 x + 6f(x)=−4x4+3x2−4x+6

Find the domain of the rational function. - g(x)=x+7x2+1g ( x ) = \frac { x + 7 } { x ^ { 2 } + 1 }g(x)=x2+1x+7​

Find the indicated intercept(s) of the graph of the function.2133:2139 - f(x)=x4−12x2−64;x−4f ( x ) = x ^ { 4 } - 12 x ^ { 2 } - 64 ; x - 4f(x)=x4−12x2−64;x−4

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not. - 7(x−1)11(x+1)37 ( x - 1 ) ^ { 11 } ( x + 1 ) ^ { 3 }7(x−1)11(x+1)3

Functions and Their Graphs

Linear and Quadratic Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometric Functions

Polar Coordinates; Vectors

Analytic Geometry

Systems of Equations and Inequalities

Sequences; Induction; the Binomial Theorem

Counting and Probability

A Preview of Calculus: the Limit, Derivative, and Integral of a Function

Review

Filters

Use Descartes' Rule of Signs and the Rational Zeros Theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the real numbers. - $f ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 9 x - 18$

Find the x- and y-intercepts of f. - $f ( x ) = x ^ { 2 } ( x - 4 ) ( x - 1 )$

Solve the problem. - $x ( x + 3 ) ( 5 - x ) \geq 0$

Find the domain of the rational function. - $f ( x ) = \frac { - 2 x ( x + 2 ) } { 2 x ^ { 2 } - 3 x - 5 }$

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not. - $f ( x ) = 5 - \frac { 3 } { x ^ { 6 } }$

Find the indicated intercept(s) of the graph of the function.2133:2139 - $y$ -intercept of $f ( x ) = \frac { x ^ { 2 } - 5 x } { x ^ { 2 } + 10 x - 13 }$

Find the x- and y-intercepts of f. - $f ( x ) = ( x + 1 ) ( x - 5 ) ( x - 1 ) ^ { 2 }$

Solve the problem. - $x ^ { 3 } + 2 x ^ { 2 } - 63 x > 0$

Solve the problem. - $x ^ { 4 } - 12 x ^ { 2 } - 64 > 0$

Find a bound on the real zeros of the polynomial function. - $f ( x ) = x ^ { 4 } + 2 x ^ { 2 } - 3$

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept. - $f ( x ) = 2 \left( x ^ { 2 } + 1 \right) \left( x ^ { 2 } + 6 \right) ^ { 2 }$

Use the Factor Theorem to determine whether x - c is a factor of f. If it is, write f in factored form, that is, write f in the form f(x) = (x - c)(quotient). - $f ( x ) = x ^ { 4 } + 10 x ^ { 3 } + 7 x ^ { 2 } + 61 x - 90 ; c = - 10$

Solve the problem. - $x ^ { 2 } - 5 x - 6 \leq 0$

List the potential rational zeros of the polynomial function. Do not find the zeros. - $f ( x ) = - 4 x ^ { 4 } + 3 x ^ { 2 } - 4 x + 6$

Find the domain of the rational function. - $g ( x ) = \frac { x + 7 } { x ^ { 2 } + 1 }$

Find the indicated intercept(s) of the graph of the function.2133:2139 - $f ( x ) = x ^ { 4 } - 12 x ^ { 2 } - 64 ; x - 4$

State whether the function is a polynomial function or not. If it is, give its degree. If it is not, tell why not. - $7 ( x - 1 ) ^ { 11 } ( x + 1 ) ^ { 3 }$