1. Section 5.1
Polynomial Functions

2. Objectives Monomials and Polynomials
Addition and Subtraction of Polynomials
Polynomial Functions
Evaluating Polynomials
Operations on Functions
Applications and Models

3. Monomials and Polynomials
A term is a number, a variable, or a product of numbers and variables raised to powers. Examples of terms: If the variables in a term have only nonnegative integer exponents, the term is called a monomial. Examples of monomials:

4. Example
Determine whether the expression is a monomial. a. b. c. d. Solution a. b. c. d.
not a monomial negative exponent
monomial
not a monomial sum (+) of two monomials
not a monomial negative exponent y-1 since 3/y = 3y-1 Division by a variable

5. Monomials
The degree of a monomial equals the sum of the exponents of the variables. A constant term has degree 0, unless the term is 0 (which as an undefined degree). The numeric constant in a monomial is called its coefficient.
The table shows the degree and coefficient of several monomials.

6. Polynomials A polynomial is either a monomial or a sum of monomials.
Polynomials containing one variable are called polynomials of one variable.
The leading coefficient of a polynomial of one variable is the coefficient of the monomial with highest degree.
The degree of a polynomial equals the degree of the monomial with the highest degree.

7. Addition and Subtraction
We can add like terms.
If two terms contain the same variables raised to the same power, we call them like terms.

8. Example
Simplify each expression by combining like terms. a. b. Solution a. b.

9. Example
Simplify the expression. Solution

10. Example
Find the sum. Solution Polynomials can be added vertically by placing like terms in the same columns and then adding each column.

11. Subtracting Polynomials
To subtract two polynomials we add the first polynomial to the opposite of the second polynomial. To find the opposite of a polynomial, we negate each term.

12. Example
Simplify. Solution The opposite of

14. Polynomial Functions
The following expressions are examples of polynomials of one variable.
As a result, we say that the following are symbolic representations of polynomial functions of one variable.

15. Example
Determine whether f(x) represents a polynomial function. If possible, identify the type of polynomial function and its degree. a. b. c.
cubic polynomial, of degree 3
not a polynomial function because the variable is negative
not a polynomial

16. Example
A graph of is shown. Evaluate f(1) graphically and check your result symbolically.
To calculate f(–1) graphically find –1 on the x-axis and move down until the graph of f is reached. Then move horizontally to the y-axis.
f(1) = –4

17. Example
Evaluate f(x) at the given value of x. Solution

19. Example
Let f(x) = 3x2 + 1 and g(x) = 6 – x2. Find each sum or difference. Solution a. b.

20. Example
Let model an athlete’s heart rate (or pulse) in beats per minute (bpm) t minutes after strenuous exercise has stopped, where 0 ≤ t ≤ 8. a. What is the initial heart rate when the athlete stops exercising?
When the athlete stops exercising, the heart rate is 200 beats per minute.

21. Example (cont)
Let model an athlete’s heart rate (or pulse) in beats per minute (bpm) t minutes after strenuous exercise has stopped, where 0 ≤ t ≤ 8. b. What is the heart rate after 8 minutes?

22. Example (cont)
Let model an athlete’s heart rate (or pulse) in beats per minute (bpm) t minutes after strenuous exercise has stopped, where 0 ≤ t ≤ 8. c. A graph of P is shown. Interpret this graph. The heart rate does not drop at a constant rate; rather, it drops rapidly at first and then gradually begins to level off.