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Section 5.1 Polynomial Functions

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**Objectives Monomials and Polynomials**

Addition and Subtraction of Polynomials Polynomial Functions Evaluating Polynomials Operations on Functions Applications and Models

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**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:

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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

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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.

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**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.

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**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.

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

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Example Simplify the expression. Solution

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

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**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.

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Example Simplify. Solution The opposite of

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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.

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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

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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

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

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

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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.

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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?

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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.

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