4-1 Polynomial Functions

A monomial is a number or a product of numbers and variables with whole number exponents. A polynomial is a monomial or a sum or difference of monomials. Each monomial in a polynomial is a term. Because a monomial has only one term, it is the simplest type of polynomial. Polynomials have no variables in denominators or exponents, no roots or absolute values of variables, and all variables have whole number exponents.

Polynomials: 3x4 2z12 + 9z3 12 a7 0.15x101 3t2 – t3

Not polynomials: 3x |2b3 – 6b| 85y2 m0.75 – m

The degree of a monomial is the sum of the exponents of the variables.

12 x

Identify the degree of each monomial.

a. x3 b. 7

c. 5x3y2 d. a6bc2

An degree of a polynomial is given by the term with the greatest degree. A polynomial with one variable is in standard form when its terms are written in descending order by degree. So, in standard form, the degree of the first term indicates the degree of the polynomial, and the leading coefficient is the coefficient of the first term.

A polynomial can be classified by its number of terms. A polynomial with two terms is called a binomial, and a polynomial with three terms is called a trinomial. A polynomial can also be classified by its degree.

Rewrite each polynomial in standard form. Then identify the leading coefficient, degree, and number of terms. Name the polynomial.

A. 3 – 5x2 + 4x B. 3x2 – 4 + 8x4

All Polynomial Functions are continuous.

A function is continuous if there are no ‘breaks’ or ‘holes’ in the graph (you can graph the function without ‘lifting your pencil’).

Which of the following could be the graph of a polynomial function?

Polynomial functions are classified by their degree. The graphs of polynomial functions are classified by the degree of the polynomial. Each graph, based on the degree, has a distinctive shape and characteristics.

End behavior is a description of the values of the function as x approaches infinity (x ∞) or negative infinity (x –∞).

The degree and leading coefficient of a polynomial function determine its end behavior.

It is helpful when you are graphing a polynomial function to know about the end behavior of the function.

For each of these functions pictured, the lead coefficient is positive. Describe the end behavior

If the lead coefficient were negative, each of the graphs would flip over the x axis. Describe the end behavior if that were the case.

Identify the leading coefficient, degree, and end behavior.

A. Q(x) = –x4 + 6x3 – x + 9

B. P(x) = 2x5 + 6x4 – x + 4

Identify whether the function graphed has an odd or even degree and a positive or negative leading coefficient.

Identify end behavior, indicate whether the function is even or odd and whether the lead coefficient is positive or negative

Identify end behavior, indicate whether the function is even or odd and whether the lead coefficient is positive or negative

A ‘root’ is a solution to a polynomial equation (sometimes called the ‘zero’ of a function).

A polynomial function is one in the form…

Where n represents a nonnegative integer.

022

110 ...)( xaxaxaxaxP n

nnn

Summary of the Fundamental Theorem of Algebra

Every polynomial of degree ‘n’ has exactly n distinct complex roots

These roots may be real or imaginary.

Real roots occur when f(x)=0, that is at the x-intercepts, and may be rational or irrational. They are easy to find on a graph using a graphing calculator.

Lesson Overview 4-1B

Polynomial functions with positive lead coefficients and maximum number of real roots.

Third and fourth degree functions with real and imaginary roots.

Describe the graph and identify the number of real zeros.

A. f(x) = 2x3 – 3x B.

Describe the graph and identify the number of real zeros.

a. f(x) = 6x3 + x2 – 5x + 1 b. f(x) = 3x2 – 2x + 2

You are familiar with factoring quadratics,

but it is sometimes possible to factor higher degree polynomials.

1662 xxxf

12112 2 xxxf

Algebraically, roots can often be found by factoring the polynomial.

xxxf 3003)( 3

1052)( 23 xxxxf

1243)( 23 xxxxf

When polynomials are presented in factored form, the zeros are easy to identify.

3 44 2f x x x

The multiplicity of root r is the number of times it occurs as a root.

When a real root has even multiplicity, the graph of y = P(x) touches the x-axis but does not cross it.

When a real root has odd multiplicity greater than 1, the graph “bends” as it crosses the x-axis.

Example: Write a 3rd degree polynomial equation of with roots 2, -1, and 4.

Example: Write a 3rd degree polynomial equation of with roots -2, multiplicity of 2, and 4, multiplicity of 1.

5-Minute Check Lesson 4-2A

Section 4-1: Polynomial Functions

Tonight’s HW: page 188-9 #37, 39, 43, 45, 47, 57-60, 73, 87, 93, 95; p191, #110