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4. Polynomial Functions

Understanding Polynomial Functions

4. Polynomial Functions

Understanding Polynomial Functions

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Introduction to Polynomial Functions

Introduction to Polynomial Functions

End Behavior of Polynomial Functions

End Behavior of Polynomial Functions

Finding Zeros & Their Multiplicity

Finding Zeros & Their Multiplicity

Maximum Turning Points of a Polynomial Function

Maximum Turning Points of a Polynomial Function

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Definition of a Polynomial Function

Definition of a Polynomial Function

Degrees and Leading Coefficients in Polynomials

Degrees and Leading Coefficients in Polynomials

What Is a Polynomial?

What Is a Polynomial?

Identifying Polynomials

Identifying Polynomials

Find Key Information about a Given Polynomial Function

Find Key Information about a Given Polynomial Function

Characteristics of Polynomial Functions

Characteristics of Polynomial Functions

End Behavior of a Polynomial Function

End Behavior of a Polynomial Function

Definition of Multiplicity of Zeros in Polynomial Functions

Definition of Multiplicity of Zeros in Polynomial Functions

Finding zeros and their multiplicities of a polynomial in factored form

Finding zeros and their multiplicities of a polynomial in factored form

How to use the Intermediate Value Theorem

How to use the Intermediate Value Theorem

Intermediate value theorem to prove a root in an interval (KristaKingMath)

Intermediate value theorem to prove a root in an interval (KristaKingMath)

Intermediate Value Theorem Examples

Intermediate Value Theorem Examples

Relative Extrema In Polynomial Functions

Relative Extrema In Polynomial Functions

Attributes of Polynomial Functions

Attributes of Polynomial Functions

Even and Odd Symmetries

Even and Odd Symmetries

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Practice this topic

Multiple Choice

Determine if the given function is a polynomial function. If so, write in standard form, then state the degree and leading coefficient. $f\left(x\right)=4x^3+\frac12x^{-1}-2x+1$

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Multiple Choice

Determine if the given function is a polynomial function. If so, write in standard form, then state the degree and leading coefficient. $f\left(x\right)=2+x$

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Multiple Choice

Determine if the given function is a polynomial function. If so, write in standard form, then state the degree and leading coefficient. $f\left(x\right)=3x^2+5x+2$

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Multiple Choice

Determine the end behavior of the given polynomial function. $f\left(x\right)=x^2+4x+x+7x^3$

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Multiple Choice

Match the given polynomial function to its graph based on end behavior. $f\left(x\right)=-2x^3+x^2+1$

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Multiple Choice

Find the zeros of the given polynomial function and give the multiplicity of each. State whether the graph crosses or touches the x-axis at each zero. $f\left(x\right)=2x^4-12x^3+18x^2$

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Multiple Choice

Find the zeros of the given polynomial function and give the multiplicity of each. State whether the graph crosses or touches the x-axis at each zero. $f\left(x\right)=x^2\left(x-1\right)^3\left(2x+6\right)$

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Multiple Choice

Determine the maximum number of turning points for the given polynomial function. $f\left(x\right)=6x^4+2x$

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Multiple Choice

Based ONLY on the maximum number of turning points, which of the following graphs could NOT be the graph of the given function? $f\left(x\right)=x^3+1$

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Multiple Choice

The given term represents the leading term of some polynomial function. Determine the end behavior and the maximum number of turning points. $4x^5$

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Textbook Question

In Exercises 1–10, determine which functions are polynomial functions. For those that are, identify the degree. f(x)=5x^2+6x^3

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Textbook Question

In Exercises 1–10, determine which functions are polynomial functions. For those that are, identify the degree. g(x)=7x^5−πx^3+1/5 x

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Textbook Question

In Exercises 1–10, determine which functions are polynomial functions. For those that are, identify the degree. h(x)=7x^3+2x^2+1/x

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Textbook Question

In Exercises 1–10, determine which functions are polynomial functions. For those that are, identify the degree. f(x)=x^1/2 −3x^2+5

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Textbook Question

In Exercises 1–10, determine which functions are polynomial functions. For those that are, identify the degree. f(x)=(x^2+7)/x^3

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Textbook Question

Graph each function. Determine the largest open intervals of the domain over which each function is (a) increasing or (b) decreasing. See Example 1. ƒ(x)=2x^4

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Textbook Question

In Exercises 1–10, determine which functions are polynomial functions. For those that are, identify the degree. f(x)=(x^2+7)/3

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Textbook Question

In Exercises 10–13, use the Leading Coefficient Test to determine the end behavior of the graph of the given polynomial function. Then use this end behavior to match the polynomial function with its graph. [The graphs are labeled (a) through (d).] f(x) = -x^3 + x^2 + 2x

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Textbook Question

In Exercises 10–13, use the Leading Coefficient Test to determine the end behavior of the graph of the given polynomial function. Then use this end behavior to match the polynomial function with its graph. [The graphs are labeled (a) through (d).] f(x) = x^6 -6x^4 + 9x^2

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Textbook Question

In Exercises 11–14, identify which graphs are not those of polynomial functions.

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Textbook Question

In Exercises 11–14, identify which graphs are not those of polynomial functions.

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Textbook Question

Graph each function. Determine the largest open intervals of the domain over which each function is (a) increasing or (b) decreasing. See Example 1. ƒ(x)=1/3(x+3)^4-3

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Textbook Question

Graph the following on the same coordinate system. (a) y = x^2 (b) y = 3x^2 (c) y = 1/3x^2 (d) How does the coefficient of x2 affect the shape of the graph?

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Textbook Question

In Exercises 19–24, (a) Use the Leading Coefficient Test to determine the graph's end behavior. (b) Determine whether the graph has y-axis symmetry, origin symmetry, or neither. (c) Graph the function. f(x) = x^3 - x^2 - 9x + 9

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Textbook Question

In Exercises 19–24, use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. f(x)=5x^3+7x^2−x+9

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Textbook Question

Graph each function. Determine the largest open intervals of the domain over which each function is (a) increasing or (b) decreasing. See Example 1. ƒ(x)=1/2(x-2)^2+4

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Textbook Question

In Exercises 19–24, (a) Use the Leading Coefficient Test to determine the graph's end behavior. (b) Determine whether the graph has y-axis symmetry, origin symmetry, or neither. (c) Graph the function. f(x) = 4x - x^3

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Textbook Question

Use an end behavior diagram, , , , or , to describe the end behavior of the graph of each polynomial function. See Example 2. ƒ(x)=5x^5+2x^3-3x+4

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Textbook Question

Use an end behavior diagram, , , , or , to describe the end behavior of the graph of each polynomial function. See Example 2. ƒ(x)=-x^3-4x^2+2x-1

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Textbook Question

In Exercises 19–24, use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. f(x)=11x^4−6x^2+x+3

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Textbook Question

In Exercises 19–24, use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. f(x)=−5x^4+7x^2−x+9

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Textbook Question

Use an end behavior diagram, , , , or , to describe the end behavior of the graph of each polynomial function. See Example 2. ƒ(x)=-4x^3+3x^2-1

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Textbook Question

Use an end behavior diagram, , , , or , to describe the end behavior of the graph of each polynomial function. See Example 2. ƒ(x)=4x^7-x^5+x^3-1

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Textbook Question

In Exercises 25–26, graph each polynomial function. f(x) = 2x^2(x - 1)^3(x + 2)

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Textbook Question

Use an end behavior diagram, , , , or , to describe the end behavior of the graph of each polynomial function. See Example 2. ƒ(x)=9x^6-3x^4+x^2-2

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Textbook Question

In Exercises 25–32, find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero. f(x)=2(x−5)(x+4)^2

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Textbook Question

In Exercises 25–32, find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero. f(x)=3(x+5)(x+2)^2

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Textbook Question

Use an end behavior diagram, , , , or , to describe the end behavior of the graph of each polynomial function. See Example 2. ƒ(x)=10x^6-x^5+2x-2

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Textbook Question

In Exercises 25–26, graph each polynomial function. f(x) = -x^3(x + 4)^2(x-1)

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Textbook Question

Use an end behavior diagram, , , , or , to describe the end behavior of the graph of each polynomial function. See Example 2. ƒ(x)=3+2x-4x^2-5x^10

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Textbook Question

In Exercises 25–32, find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero. f(x)=−3(x+1/2)(x−4)^3

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Textbook Question

Use an end behavior diagram, , , , or , to describe the end behavior of the graph of each polynomial function. See Example 2. ƒ(x)=7+2x-5x^2-10x^4

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Textbook Question

In Exercises 25–32, find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero. f(x)=x^3+7x^2−4x−28

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Textbook Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4. ƒ(x)=-2x(x-3)(x+2)

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Textbook Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4. ƒ(x)=-x(x+1)(x-1)

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Textbook Question

In Exercises 33–40, use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=x^3−x−1; between 1 and 2

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Textbook Question

In Exercises 33–40, use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=x^3−4x^2+2; between 0 and 1

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Textbook Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4. ƒ(x)=(3x-1)(x+2)^2

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Textbook Question

In Exercises 33–40, use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=x^4+6x^3−18x^2; between 2 and 3

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Textbook Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4. ƒ(x)=(4x+3)(x+2)^2

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Textbook Question

Determine the largest open interval of the domain (a) over which the function is increasing and (b) over which it is decreasing. See Example 2. ƒ(x) = (x + 3)^2

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Textbook Question

In Exercises 33–40, use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=x^3+x^2−2x+1; between -3 and -2

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Textbook Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4. ƒ(x)=x^3+5x^2-x-5

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Textbook Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4. ƒ(x)=x^3+x^2-36x-36

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Textbook Question

Determine the largest open interval of the domain (a) over which the function is increasing and (b) over which it is decreasing. See Example 2. ƒ(x) = -(x - 2)^2 - 5

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Textbook Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4. ƒ(x)=-x^3+x^2+2x

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Textbook Question

In Exercises 33–40, use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=3x^3−8x^2+x+2; between 2 and 3

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Textbook Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4. ƒ(x)=-3x^4-5x^3+2x^2

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Textbook Question

Determine the largest open interval of the domain (a) over which the function is increasing and (b) over which it is decreasing. See Example 2. ƒ(x) = x^2 - 4x + 3

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Textbook Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4. ƒ(x)=2x^3(x^2-4)(x-1)

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Textbook Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4. ƒ(x)=2x^3-5x^2-x+6

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Textbook Question

Determine the largest open interval of the domain (a) over which the function is increasing and (b) over which it is decreasing. See Example 2. ƒ(x) = -2x^2 - 8x - 7

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Textbook Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4. ƒ(x)=2x^4+x^3-6x^2-7x-2

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Textbook Question

Determine the largest open interval of the domain (a) over which the function is increasing and (b) over which it is decreasing. See Example 2. ƒ(x) = -3x^2 + 18x + 1

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Textbook Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4. ƒ(x)=3x^4-7x^3-6x^2+12x+8

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Textbook Question

If the given term is the dominating term of a polynomial function, what can we conclude about each of the following features of the graph of the function? (a)domain (b)range (c)end behavior (d)number of zeros (e)number of turning points 10x7

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Textbook Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. See Examples 3 and 4. ƒ(x)=x^4+3x^3-3x^2-11x-6

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Textbook Question

If the given term is the dominating term of a polynomial function, what can we conclude about each of the following features of the graph of the function? (a)domain (b)range (c)end behavior (d)number of zeros (e)number of turning points -9x6

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Textbook Question

Graph each polynomial function. ƒ(x)=(x-2)^2(x+3)

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Textbook Question

Use the intermediate value theorem to show that each polynomial function has a real zero between the numbers given. See Example 5. ƒ(x)=3x^2-x-4; 1 and 2

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Textbook Question

Graph each polynomial function. ƒ(x)=2x^3+x^2-x

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Textbook Question

Use the intermediate value theorem to show that each polynomial function has a real zero between the numbers given. See Example 5. ƒ(x)=-2x^3+5x^2+5x-7; 0 and 1

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Textbook Question

Use the intermediate value theorem to show that each polynomial function has a real zero between the numbers given. See Example 5. ƒ(x)=2x^4-4x^2+4x-8; 1 and 2

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Textbook Question

Graph each polynomial function. ƒ(x)=-2x^4+7x^3-4x^2-4x

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Textbook Question

Use the intermediate value theorem to show that each polynomial function has a real zero between the numbers given. See Example 5. ƒ(x)=x^4-4x^3-x+3; 0.5 and 1

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Textbook Question

For each polynomial function, identify its graph from choices A–F. ƒ(x)=(x-2)^2(x-5)

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Textbook Question

For each polynomial function, identify its graph from choices A–F. ƒ(x)=-(x-2)^2(x-5)

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Textbook Question

For each polynomial function, identify its graph from choices A–F. ƒ(x)=(x-2)^2(x-5)^2

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Textbook Question

For each polynomial function, identify its graph from choices A–F. ƒ(x)=(x-2)(x-5)

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Textbook Question

Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6. ƒ(x)=x^4-x^3+3x^2-8x+8; no real zero greater than 2

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Textbook Question

For each polynomial function, identify its graph from choices A–F. ƒ(x)=-(x-2)(x-5)

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Textbook Question

Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6. ƒ(x)=2x^5-x^4+2x^3-2x^2+4x-4; no real zero greater than 1

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Textbook Question

For each polynomial function, identify its graph from choices A–F. ƒ(x)=-(x-2)^2(x-5)^2

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Textbook Question

Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6. ƒ(x)=x^4+x^3-x^2+3; no real zero less than -2

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Textbook Question

Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6. ƒ(x)=x^5+2x^3-2x^2+5x+5; no real zero less than -1

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Textbook Question

Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6. ƒ(x)=3x^4+2x^3-4x^2+x-1; no real zero greater than 1

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Textbook Question

Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6. ƒ(x)=3x^4+2x^3-4x^2+x-1; no real zero less than -2

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Textbook Question

Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6. ƒ(x)=x^5-3x^3+x+2; no real zero greater than 2

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Textbook Question

Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6. ƒ(x)=x^5-3x^3+x+2; no real zero less than -3

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Textbook Question

Find a polynomial function f of least degree having the graph shown. (Hint: See the NOTE following Example 4.)

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Textbook Question

Find a polynomial function f of least degree having the graph shown. (Hint: See the NOTE following Example 4.)

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Textbook Question

Use a graphing calculator to find the coordinates of the turning points of the graph of each polynomial function in the given domain interval. Give answers to the nearest hundredth. ƒ(x)=2x^3-5x^2-x+1; [-1, 0]

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Textbook Question

Use a graphing calculator to find the coordinates of the turning points of the graph of each polynomial function in the given domain interval. Give answers to the nearest hundredth. ƒ(x)=2x^3-5x^2-x+1; [1.4, 2]

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Textbook Question

Use a graphing calculator to find the coordinates of the turning points of the graph of each polynomial function in the given domain interval. Give answers to the nearest hundredth. ƒ(x)=x^3+4x^2-8x-8; [-3.8, -3]

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Textbook Question

Use a graphing calculator to find the coordinates of the turning points of the graph of each polynomial function in the given domain interval. Give answers to the nearest hundredth. ƒ(x)=x^4-7x^3+13x^2+6x-28; [-1, 0]

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Textbook Question

The following exercises are geometric in nature and lead to polynomial models. Solve each problem. A standard piece of notebook paper measuring 8.5 in. by 11 in. is to be made into a box with an open top by cutting equal-size squares from each cor-ner and folding up the sides. Let x represent the length of a side of each such square in inches. Use the table feature of a graphing calculator to do the following. Round to the nearest hundredth. Determine when the volume of the box will be greater than 40 in.^3.

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Textbook Question

The following exercises are geometric in nature and lead to polynomial models. Solve each problem. A standard piece of notebook paper measuring 8.5 in. by 11 in. is to be made into a box with an open top by cutting equal-size squares from each cor-ner and folding up the sides. Let x represent the length of a side of each such square in inches. Use the table feature of a graphing calculator to do the following. Round to the nearest hundredth. Find the maximum volume of the box.

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Textbook Question

Exercises 107–109 will help you prepare for the material covered in the next section. Factor: x^3+3x^2−x−3

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Textbook Question

Exercises 107–109 will help you prepare for the material covered in the next section. Determine whether f(x)=x^4−2x^2+1 is even, odd, or neither. Describe the symmetry, if any, for the graph of f.

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Textbook Question

Rewrite 4-5x-x^2+6x^3 in descending powers of x.

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Textbook Question

Use (2x^3−3x^2−11x+6)/(x−3)=2x^2+3x−2 to factor 2x^3-3x^2-11x+6 completely.

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