Related Topics:

Remainder Theorem

Common Core (Algebra)

Common Core for Mathematics

### Suggested Learning Targets

**Real Zeros, Factors, and Graphs of Polynomial Functions**

This video explains the connection between zero, factors, and graphs of polynomial functions.

Assume we have a polynomial function of degree n

P(x) = a_{n}x^{n} + a_{n-1}x^{n-1} + ... + a_{1}x + a_{0}

and c is a real number such that P(c) = 0.

1. The real number c is a zero or root of P(x).

2. x = c is a solution to the equation p(x) = 0.

3. (x - c) is a factor of P(x).

4. The point (c, 0) is an x-intercept of the graph of P(x).

Examples:

1. A degree 4 polynomial has zeros or roots of multiplicity 1 at x = 0 and x = 3 and a zero of multiplicity of 2 at x = -1 and has a leading coefficient of -2.

a. Give the x-intercepts of the polynomial function.

b. Write the polynomial function in factored form.

c. Graph the polynomial function.

2. Given the graph of a degree 3 polynomial.

a. Find the x-intercepts.

b. Find the real zeros or roots of the function.

c. Find the equation of the polynomial function.

**Sketch the graph of polynomials using zeros, end behavior, and y intercept**

Example:

Sketch the graphs of the following polynomials.

1. g(x) = x^{4} - 4x^{3} + 3x^{2}

2. f(x) = -6x^{3} + 8x^{2} + 54x - 72

3. h(x) = -2x(x-3)^{3}(x+5)^{2}

**End Behavior of Polynomial Functions**

What is 'End Behavior'? How do I describe the end behavior of a polynomial function?

Examples:

Find the end behavior of the following polynomial function.

a) f(x) = x^{4} + 2x^{3} + 8

b) f(x) = 3x^{5} + 2x^{4} - 3x + 5
**Graphing Polynomials Part 1**
**Graphing Polynomials Part 2**

How the degree and multiplicity affect the graph of a polynomial.**Finding Zeros and Factors of Polynomial Functions**

Find all zeros of a polynomial function. This video covers many examples using factoring, graphing, and synthetic division.

Examples:

Find the zeros and factors of the following functions:

1. f(x) = x^{2} + 7x + 12

2. f(x) = x^{3} + x^{2} - 9x - 9

3. f(x) = x^{3} + x^{2} - 14x - 24

4. f(x) = x^{3} + 5x^{2} + 3x - 6

5. f(x) = x^{4} + x^{3} - 19x^{2} - 19x + 60

5. f(x) = x^{4} + 3x^{3} + 14x^{2} + 3x + 1

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

Remainder Theorem

Common Core (Algebra)

Common Core for Mathematics

Videos and lessons to help High School students identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

- Find the zeros of a polynomial when the polynomial is factored.
- Use the zeros of a function to sketch a graph of the function.

Common Core: HSA-APR.B.3

The following figure show how to find the zeros or roots of a polynomial function. Scroll down the page for more examples and solutions.This video explains the connection between zero, factors, and graphs of polynomial functions.

Assume we have a polynomial function of degree n

P(x) = a

and c is a real number such that P(c) = 0.

1. The real number c is a zero or root of P(x).

2. x = c is a solution to the equation p(x) = 0.

3. (x - c) is a factor of P(x).

4. The point (c, 0) is an x-intercept of the graph of P(x).

Examples:

1. A degree 4 polynomial has zeros or roots of multiplicity 1 at x = 0 and x = 3 and a zero of multiplicity of 2 at x = -1 and has a leading coefficient of -2.

a. Give the x-intercepts of the polynomial function.

b. Write the polynomial function in factored form.

c. Graph the polynomial function.

2. Given the graph of a degree 3 polynomial.

a. Find the x-intercepts.

b. Find the real zeros or roots of the function.

c. Find the equation of the polynomial function.

Example:

Sketch the graphs of the following polynomials.

1. g(x) = x

2. f(x) = -6x

3. h(x) = -2x(x-3)

What is 'End Behavior'? How do I describe the end behavior of a polynomial function?

Examples:

Find the end behavior of the following polynomial function.

a) f(x) = x

b) f(x) = 3x

How the degree and multiplicity affect the graph of a polynomial.

Find all zeros of a polynomial function. This video covers many examples using factoring, graphing, and synthetic division.

Examples:

Find the zeros and factors of the following functions:

1. f(x) = x

2. f(x) = x

3. f(x) = x

4. f(x) = x

5. f(x) = x

5. f(x) = x

Rotate to landscape screen format on a mobile phone or small tablet to use the **Mathway** widget, a free math problem solver that **answers your questions with step-by-step explanations**.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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