Exam 3: Polynomial and Rational Functions

Determine the maximum number of turning points of f. - $f ( x ) = x ^ { 2 } ( x + 2 )$ Answer: (a) For large values of $| x |$ , the graph of $f ( x )$ will resemble the graph of $y = x ^ { 3 }$ . (b) $y$ -intercept: $( 0,0 )$ , $x$ -intercepts: $( 0,0 )$ and $( - 2,0 )$ (c) The graph of $f$ crosses the $x$ -axis at $( - 2,0 )$ and touches the $x$ -axis at $( 0,0 )$ . (e) Local minimum at $( 0,0 )$ , Local maximum at $( - 1.33,1.19 )$ (f) $Determine the maximum number of turning points of f. - f ( x ) = x ^ { 2 } ( x + 2 ) Answer: (a) For large values of | x | , the graph of f ( x ) will resemble the graph of y = x ^ { 3 } . (b) y -intercept: ( 0,0 ) , x -intercepts: ( 0,0 ) and ( - 2,0 ) (c) The graph of f crosses the x -axis at ( - 2,0 ) and touches the x -axis at ( 0,0 ) . (e) Local minimum at ( 0,0 ) , Local maximum at ( - 1.33,1.19 ) (f) (g) Domain of f: all real numbers; range of f: all real numbers (h) f is increasing on (-, -1.33) and (0, ); f is decreasing on (-1.33, 0)$ (g) Domain of f: all real numbers; range of f: all real numbers (h) f is increasing on (-, -1.33) and (0, ); f is decreasing on (-1.33, 0)

(Short Answer)

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Question 81

Use the Intermediate Value Theorem to determine whether the polynomial function has a zero in the given interval. -f(x) = x⁵ - 10x⁴ + 42x³ -124 x² + 297x - 306 ; zero: 3i

(Multiple Choice)

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Question 82

Find the x- and y-intercepts of f. - $f ( x ) = ( x + 6 ) ^ { 2 }$

(Multiple Choice)

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Question 83

Find the x- and y-intercepts of f. - $f ( x ) = 2 x ^ { 4 } ( x - 5 ) ^ { 5 }$

(Multiple Choice)

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Question 84

$f ( x ) = 5 \left( x ^ { 2 } + 6 \right) \left( x ^ { 2 } + 4 \right) ^ { 2 }$

(Multiple Choice)

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Question 85

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) = 19x⁵ + 9x⁴ + 2

(Multiple Choice)

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Question 86

Find the x- and y-intercepts of f. - $f ( x ) = - x ^ { 2 } ( x + 6 ) \left( x ^ { 2 } + 1 \right)$ $( x ) = - x ^ { 2 } ( x + 6 ) \left( x ^ { 2 } + 1 \right)$

(Multiple Choice)

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Question 87

Solve the equation in the real number system. - $x ^ { 4 } - 3 x ^ { 3 } + 5 x ^ { 2 } - x - 10 = 0$

(Multiple Choice)

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Question 88

Solve the equation in the real number system. - $x ^ { 3 } + 3 x ^ { 2 } - 10 x - 24 = 0$

(Multiple Choice)

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Question 89

Find the x- and y-intercepts of f. -f(x) = (x - 2)(x - 5)

(Multiple Choice)

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Question 90

(Multiple Choice)

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Question 91

Use the Intermediate Value Theorem to determine whether the polynomial function has a zero in the given interval. -f(x) = 3x³ - 6x + 2; [1, 2]

(Multiple Choice)

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Question 92

The equation has a solution r in the interval indicated. Approximate this solution correct to two decimal places. -x³ - 8x - 3 = 0; -1 ≤ r ≤ 0

(Short Answer)

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Question 93

Determine the maximum number of turning points of f. -f(x) = 8x - x³

(Multiple Choice)

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Question 94

The equation has a solution r in the interval indicated. Approximate this solution correct to two decimal places. -x⁴ - x³ - 7x² + 5x + 10 = 0; 2 < r ≤ 3

(Short Answer)

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Question 95

Find the domain of the rational function. - $g ( x ) = \frac { 6 x } { x - 3 }$

(Multiple Choice)

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Question 96

Showing 81 - 96 of 96

Use the Intermediate Value Theorem to determine whether the polynomial function has a zero in the given interval. -f(x) = x⁵ - 10x⁴ + 42x³ -124 x² + 297x - 306 ; zero: 3i

Find the x- and y-intercepts of f. - $f ( x ) = ( x + 6 ) ^ { 2 }$

Find the x- and y-intercepts of f. - $f ( x ) = 2 x ^ { 4 } ( x - 5 ) ^ { 5 }$

$f ( x ) = 5 \left( x ^ { 2 } + 6 \right) \left( x ^ { 2 } + 4 \right) ^ { 2 }$

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) = 19x⁵ + 9x⁴ + 2

Find the x- and y-intercepts of f. - $f ( x ) = - x ^ { 2 } ( x + 6 ) \left( x ^ { 2 } + 1 \right)$ $( x ) = - x ^ { 2 } ( x + 6 ) \left( x ^ { 2 } + 1 \right)$

Solve the equation in the real number system. - $x ^ { 4 } - 3 x ^ { 3 } + 5 x ^ { 2 } - x - 10 = 0$

Solve the equation in the real number system. - $x ^ { 3 } + 3 x ^ { 2 } - 10 x - 24 = 0$

Find the x- and y-intercepts of f. -f(x) = (x - 2)(x - 5)

Use the Intermediate Value Theorem to determine whether the polynomial function has a zero in the given interval. -f(x) = 3x³ - 6x + 2; [1, 2]

The equation has a solution r in the interval indicated. Approximate this solution correct to two decimal places. -x³ - 8x - 3 = 0; -1 ≤ r ≤ 0

Determine the maximum number of turning points of f. -f(x) = 8x - x³

The equation has a solution r in the interval indicated. Approximate this solution correct to two decimal places. -x⁴ - x³ - 7x² + 5x + 10 = 0; 2 < r ≤ 3

Find the domain of the rational function. - $g ( x ) = \frac { 6 x } { x - 3 }$

Functions and Their Graphs

Linear and Quadratic Functions

Conics

Systems of Equations and Inequalities

Sequences; Induction; the Binomial Theorem

Counting and Probability

Filters

Exam 3: Polynomial and Rational Functions

Use the Intermediate Value Theorem to determine whether the polynomial function has a zero in the given interval. -f(x) = x5 - 10x4 + 42x3 -124 x2 + 297x - 306 ; zero: 3i

Find the x- and y-intercepts of f. - f(x)=(x+6)2f ( x ) = ( x + 6 ) ^ { 2 }f(x)=(x+6)2

Find the x- and y-intercepts of f. - f(x)=2x4(x−5)5f ( x ) = 2 x ^ { 4 } ( x - 5 ) ^ { 5 }f(x)=2x4(x−5)5

f(x)=5(x2+6)(x2+4)2f ( x ) = 5 \left( x ^ { 2 } + 6 \right) \left( x ^ { 2 } + 4 \right) ^ { 2 }f(x)=5(x2+6)(x2+4)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) = 19x5 + 9x4 + 2

Find the x- and y-intercepts of f. - f(x)=−x2(x+6)(x2+1)f ( x ) = - x ^ { 2 } ( x + 6 ) \left( x ^ { 2 } + 1 \right)f(x)=−x2(x+6)(x2+1) (x)=−x2(x+6)(x2+1)( x ) = - x ^ { 2 } ( x + 6 ) \left( x ^ { 2 } + 1 \right)(x)=−x2(x+6)(x2+1)

Solve the equation in the real number system. - x4−3x3+5x2−x−10=0x ^ { 4 } - 3 x ^ { 3 } + 5 x ^ { 2 } - x - 10 = 0x4−3x3+5x2−x−10=0

Solve the equation in the real number system. - x3+3x2−10x−24=0x ^ { 3 } + 3 x ^ { 2 } - 10 x - 24 = 0x3+3x2−10x−24=0

Find the x- and y-intercepts of f. -f(x) = (x - 2)(x - 5)

Use the Intermediate Value Theorem to determine whether the polynomial function has a zero in the given interval. -f(x) = 3x3 - 6x + 2; [1, 2]

The equation has a solution r in the interval indicated. Approximate this solution correct to two decimal places. -x3 - 8x - 3 = 0; -1 ≤ r ≤ 0

Determine the maximum number of turning points of f. -f(x) = 8x - x3

The equation has a solution r in the interval indicated. Approximate this solution correct to two decimal places. -x4 - x3 - 7x2 + 5x + 10 = 0; 2 < r ≤ 3

Find the domain of the rational function. - g(x)=6xx−3g ( x ) = \frac { 6 x } { x - 3 }g(x)=x−36x​

Functions and Their Graphs

Linear and Quadratic Functions

Conics

Systems of Equations and Inequalities

Sequences; Induction; the Binomial Theorem

Counting and Probability

Filters

Use the Intermediate Value Theorem to determine whether the polynomial function has a zero in the given interval. -f(x) = x⁵ - 10x⁴ + 42x³ -124 x² + 297x - 306 ; zero: 3i

Find the x- and y-intercepts of f. - $f ( x ) = ( x + 6 ) ^ { 2 }$

Find the x- and y-intercepts of f. - $f ( x ) = 2 x ^ { 4 } ( x - 5 ) ^ { 5 }$

$f ( x ) = 5 \left( x ^ { 2 } + 6 \right) \left( x ^ { 2 } + 4 \right) ^ { 2 }$

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) = 19x⁵ + 9x⁴ + 2

Find the x- and y-intercepts of f. - $f ( x ) = - x ^ { 2 } ( x + 6 ) \left( x ^ { 2 } + 1 \right)$ $( x ) = - x ^ { 2 } ( x + 6 ) \left( x ^ { 2 } + 1 \right)$

Solve the equation in the real number system. - $x ^ { 4 } - 3 x ^ { 3 } + 5 x ^ { 2 } - x - 10 = 0$

Solve the equation in the real number system. - $x ^ { 3 } + 3 x ^ { 2 } - 10 x - 24 = 0$

Use the Intermediate Value Theorem to determine whether the polynomial function has a zero in the given interval. -f(x) = 3x³ - 6x + 2; [1, 2]

The equation has a solution r in the interval indicated. Approximate this solution correct to two decimal places. -x³ - 8x - 3 = 0; -1 ≤ r ≤ 0

Determine the maximum number of turning points of f. -f(x) = 8x - x³

The equation has a solution r in the interval indicated. Approximate this solution correct to two decimal places. -x⁴ - x³ - 7x² + 5x + 10 = 0; 2 < r ≤ 3

Find the domain of the rational function. - $g ( x ) = \frac { 6 x } { x - 3 }$