4. Polynomial and Rational Functions

Use the Intermediate Value Theorem

Polynomial Functions and Their Graphs

Intermediate Value Theorem

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

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

Use Factoring to Find Zeros of Polynomial Functions

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

Determine when a polynomial function changes sign

Finding all the Zeros of a Polynomial - Example 2

 Definition of a Polynomial Function

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

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

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

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

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

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

In Exercises 11–14, identify which graphs are not those of polynomial functions. <p><img src="https://lightcat-files.s3.amazonaws.com/problem_images/0ecd5f68d0a27aa0-1674234483737.jpg" alt="" /></p>

Finding Maximum Number of Turning Points

Finding zeros and their multiplicities of a polynomial in factored form

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

How to use the Intermediate Value Theorem

Identify polynomial extrema

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.

Characteristics of Polynomial Functions - with graph

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

Local and Absolute Maximum and Minimum from a Graph

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

Polynomial... or NOT?! Recognizing Polynomials, the degree and some Terminology

How to Find all Zeros of Polynomial

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

Determine the end behavior of a polynomial function

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

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

Finding all the Zeros of a Polynomial - Example 3

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

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

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

 End Behavior of a Polynomial Function

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

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

Determine end behavior of polynomials by exploring patterns

Definition of Multiplicity of Zeros in Polynomial Functions

Finding Critical Numbers 

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

In Exercises 11–14, identify which graphs are not those of polynomial functions. <p><img src="https://lightcat-files.s3.amazonaws.com/problem_images/d8b8a5b1c61b500b-1674232322826.jpg" alt="" /></p>

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

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

Determining the Zeros (and Their Multiplicity) of a Polynomial Graphically

 Identifying Polynomials

Relative Extrema In Polynomial Functions

Characteristics of Polynomial Functions

Determine polynomial end behavior

 What Is a Polynomial?

Graphing Higher-Degree Polynomials: The Leading Coefficient Test and Finding Zeros

 Even and Odd Symmetries

 Degrees and Leading Coefficients in Polynomials

Graph a polynomial by finding the x-intercepts

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

Intermediate Value Theorem Examples

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

Factor Polynomial Given a Complex / Imaginary Root

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

Finding all the Zeros of a Polynomial - Example 1

Attributes of Polynomial Functions

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

Sketch the graph of a higher degree polynomial

Graph polynomials by finding the zeros through factoring

Find Key Information about a Given Polynomial Function

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

Learn the toughest concepts covered in Algebra with step-by-step video tutorials and practice problems by world-class tutors

College Algebra

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1. Fundamental Concepts of Algebra

2. Equations and Inequalities

3. Functions and Graphs

5. Exponential and Logarithmic Functions

6. Systems of Equations and Inequalities

7. Matrices and Determinants

8. Conic Sections

9. Sequences, Induction, and Probability

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Use Long Division to Divide Polynomials

Use Synthetic Division to Divide Polynomials

Evaluate a Polynomial Using the Remainder Theorem

Use the Factor Theorem to Solve a Polynomial Equation

Dividing Polynomials

Identify Polynomial Functions

Recognize Characteristics of Graphs of Polynomial Functions

Determine End Behavior

Identify Zeros and Their Multiplicities

Understand the Relationship Between Degree and Turning Points

Graph Polynomial Functions

Solve Polynomial Inequalities

Solve Rational Inequalities

 Problems Modeled by Polynomial or Rational Inequalities

Polynomial and Rational Inequalities

Find the Domains of Rational Functions

Use Arrow Notation

Identify Vertical Asymptotes

Identify Horizontal Asymptotes

Use Transformations to Graph Rational Functions

Graph Rational Functions

Identify Slant Asymptotes

Solve Applied Problems Involving Rational Functions

Rational Functions and Their Graphs

Recognize Characteristics of Parabolas

Graph Parabolas

Determine a Quadratic Function's Minimum or Maximum

Problems Involving a Quadratic Function's Minimum or Maximum

Quadratic Functions

Solve Direct Variation Problems

Solve Inverse Variation Problems

Solve Combined Variation Problems

Solve Problems Involving Joint Variation

Modeling Using Variation

Use Rational Zero Theorem to Find Possible Rational Zeros

Find Zeros of a Polynomial Function

Solve Polynomial Equations

Use the Linear Factorization Theorem to Find Polynomials With Given Zeros

Use Descartes's Rule of Signs

Zeros of Polynomial Functions

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