1. 4-1 Polynomial Functions

2. The degree of a monomial is the sum of the exponents of the variables.
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. 1 2 a7
Polynomials:
3x4
2z12 + 9z3
0.15x101
3t2 – t3 8
5y2 1 2
Not polynomials: 3x
|2b3 – 6b|
m0.75 – m
The degree of a monomial is the sum of the exponents of the variables.

3. Identify the degree of each monomial.
a. x3
b. 7
c. 5x3y2
d. a6bc2

4. 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.

5. A polynomial can be classified by its number of terms
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.

6. Rewrite each polynomial in standard form
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

7. 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’).
All Polynomial Functions are continuous.

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

10. Polynomial functions are classified by their degree
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.

11. 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.

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

14. 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.

15. Identify the leading coefficient, degree, and end behavior.
A. Q(x) = –x4 + 6x3 – x + 9
B. P(x) = 2x5 + 6x4 – x + 4

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

17. 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

18. A polynomial function is one in the form…
Where n represents a nonnegative integer.
A ‘root’ is a solution to a polynomial equation (sometimes called the ‘zero’ of a function).

19. 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.

20. 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.

23. B. Describe the graph and identify the number of real zeros.
A. f(x) = 2x3 – 3x

24. Describe the graph and identify the number of real zeros.
a. f(x) = 6x3 + x2 – 5x + 1
b. f(x) = 3x2 – 2x + 2

25. Algebraically, roots can often be found by factoring the polynomial.
You are familiar with factoring quadratics,
but it is sometimes possible to factor higher degree polynomials.

27. The multiplicity of root r is the number of times it occurs as a root.
When polynomials are presented in factored form, the zeros are easy to identify.
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.

28. 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.

29. 5-Minute Check Lesson 4-2A

30. Section 4-1: Polynomial Functions
Tonight’s HW: page #37, 39, 43, 45, 47, 57-60, 73, 87, 93, 95; p191, #110