2 Objectives Monomials and Polynomials Addition and Subtraction of PolynomialsPolynomial FunctionsEvaluating PolynomialsOperations on FunctionsApplications 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 ExampleDetermine whether the expression is a monomial. a. b. c. d. Solution a. b. c. d.not a monomial negative exponentmonomialnot a monomial sum (+) of two monomialsnot a monomial negative exponent y-1 since 3/y = 3y-1 Division by a variable
5 MonomialsThe 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 ExampleSimplify each expression by combining like terms. a. b. Solution a. b.
14 Polynomial FunctionsThe 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 ExampleDetermine 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 3not a polynomial function because the variable is negativenot a polynomial
16 ExampleA 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 ExampleEvaluate f(x) at the given value of x. Solution
19 ExampleLet f(x) = 3x2 + 1 and g(x) = 6 – x2. Find each sum or difference. Solution a. b.
20 ExampleLet 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.