Algebra 2A, Chapter 5 Notes, Part 1 1

Algebra 2A, Chapter 5 Notes, Part 1 1

5.1 Polynomial Functions

Terms

Monomial:

Degree of a Monomial:

Polynomial (in one variable):

Degree of a Polynomial:

Polynomial Function:

You can classify a polynomial l by its degree or its number of terms.

Polynomials degree 0-5 have specific names.

Degree: Number of terms:

0constant1monomial

1linear2binomial

2quadratic3trinomial

3cubicNPolynomial of “n” terms

4quartic

5quintic

Examples:

Example 1: Classifying Polynomials

Write each polynomial in standard form.

What is its classification by degree? By number of terms?

A.D.

B.E.

C. F.

Degree of a Polynomial Function

• Affects the shape of the graph

• Affects the end behavior
• End behavior:

4 Types of End Behavior (for functions of degree one or greater)

You can determine the end behavior of a polynomial function of degree n from the leading term axn of standard form.

Example 2: Describing the End Behavior of Polynomial Functions

Consider the leading term of each polynomial function. What is its end behavior?

A.B.

C.D.

E.F.

Example 3: Graphing a Cubic Function

Graph:

5.2 Polynomials, Linear Factors and Zeros

If P(x) is a polynomial function, the solutions of the related polynomial equation P(x) = 0 are the zeros of the function.

Example:

Polynomial Function

Polynomial Equation:

Zeros:

Finding the zeros of a polynomial function will help you factor the polynomial, graph the function, and solve the related polynomial equation.

Key Concept: Roots, Zeros and X-Intercepts

The following are equivalent statements about a real number b and a polynomial

• is a linear factor of the polynomial P(x)
• b is a zero of the polynomial function
• b is a root (or solution) of the polynomial equation
• b is an x-intercept of the graph of

Example: Let and let x = 2.Then P(2) = 0

Therefore,

• ______ is a factor of

• ______is a zero for the polynomial function P(x)

• ______is a solution to the equation

• The point (____,____)is an x-intercept of the graph of

Review: Factoring

Factor the following polynomials completely.

A.B.C.

Example 1: Finding Zeros of a Polynomial Function

A) What are the zeros of ? Sketch the graph of the function.

B) What are the zeros of ? Sketch the graph of the function.

Theorem: Factor Theorem

The expression is a factor of the polynomial if and only if the value a is a zero of the related function.

Example 2: Writing a Polynomial From Its Zeros

A.What is a polynomial function with zeros 4 and -4?

Write a polynomial function with the given zeros.

B.3, 3, and -4C. 2, 1, and -1

Key Concept: Multiplicity

can be written as .

Because the linear factor appears twice, -2 is a zero of multiplicity 2.

In general, a is a zero of multiplicity n means that appears n times as a factor of the polynomial.

Example 3: Finding the Multiplicity of a Zero

Find the zeros of the function. State the multiplicity of any multiple zeros.

A) B)

C) D)

5.3 Solving Polynomial Equations

Recall: The expression is a factor of the polynomial if and only if the value a is a zero of the related function.

To solve a polynomial equation by factoring,

1. Write the equation in the form P(x) = 0for the polynomial function P.
2. Factor P(x). Use the Zero Product Property to find the roots.

Practice:

Factor:Factor:

Solving Polynomial Equations by Factoring

What are the real or imaginary solutions of each polynomial equation?

1.2.

3.4.

5.6.

7.8.

5.4 Dividing Polynomials

You can divide polynomials using steps similar to the long division steps used to divide whole numbers.

Example 1: Using Polynomial Long Division

A)Divide by .

B)Divide by .

Key Concept: Synthetic Division

Synthetic division simplifies the long division process for dividing by a linear expression .

To use synthetic division,

1. Write the coefficients (including zeros) of the polynomial in standard form.

2. For the divisor, reverse the sign (use a).

Example 2: Using Synthetic Division

A)Use synthetic division to divide by .

B)Use synthetic division to divide by .

C)Use synthetic division to divide by .

Example 3: Checking Factors

is factor of the polynomial function if and only if has a remainder of zero.

A) Is a factor of ? Why or why not?

B) Is a factor of ? Why or why not?

Key Concept: Remainder Theorem

If you divide a polynomial of degree by , then the remainder is

Example 4: Evaluating a Polynomial Using Synthetic Division

Sometimes this is called “Synthetic Substitution”

Use synthetic division and the remainder theorem to find the following:

A) for

B) for

C) for