190 CHAPTER 4 Polynomial and Rational Functions 4.1 Polynomial Functions Now Work the ‘Are You Prepared?’ problems on page 201. • Polynomials (Section A.3, pp. A22–A29) • Obtaining Information from or about the Graph of a Function (Section 2.2, pp. 77–81) • Graphing Techniques: Transformations (Section 2.5, pp. 112–120) • Intercepts (Section 1.3, pp. 20–21) • Library of Functions (Section 2.4, pp. 100–105) PREPARING FOR THIS SECTION Before getting started, review the following: Mary Golda Ross (1908–2008) Mary Golda Ross was a mathematician and engineer and was a member of the Cherokee Nation. She received her bachelor’s degree and a master's degree in mathematics in 1928 and 1938 respectively. She went on to work for Lockheed as a mathematician in 1941. Ross later helped author the NASA Planetary Flight Handbook Vol. III. In Words A polynomial function is a sum of monomials. Identifying Polynomial Functions Determine which of the following are polynomial functions. For those that are, state the degree; for those that are not, state why not. Write each polynomial function in standard form, and then identify the leading term and the constant term. (a) p x x x 5 1 4 9 3 2 ( ) = − − (b) ( ) = + − f x x x 2 3 4 (c) g x x ( ) = (d) h x x x 2 1 2 3 ( ) = − − (e) G x 8 ( ) = (f) ( ) ( ) =− − H x x x 2 1 3 2 EXAMPLE 1 1 Identify Polynomial Functions and Their Degree In Chapter 3, we studied the linear function f x mx b, ( ) = + which can be written as f x a x a 1 0 ( ) = + and the quadratic function f x ax bx c a , 0, 2 ( ) = + + ≠ which can be written as f x a x a x a a 0 2 2 1 0 2 ( ) = + + ≠ Both of these functions are examples of a polynomial function . OBJECTIVES 1 Identify Polynomial Functions and Their Degree (p. 190) 2 Graph Polynomial Functions Using Transformations (p. 194) 3 Identify the Real Zeros of a Polynomial Function and Their Multiplicity (p. 194) DEFINITION Polynomial Function A polynomial function in one variable is a function of the form f x a x a x a x a n n n n 1 1 1 0 ( ) = + + + + − − (1) where … a a a a , ,, , n n 1 1 0 − are constants, called the coefficients of the polynomial; n 0 ≥ is an integer; and x is a variable. If a 0, n ≠ it is called the leading coefficient , and n is the degree of the polynomial. The domain of a polynomial function is the set of all real numbers. The monomials that make up a polynomial function are called its terms . If a a x 0, n n n ≠ is called the leading term ; a 0 is called the constant term . If all of the coefficients are 0, the polynomial is called the zero polynomial , which has no degree. Polynomial functions are usually written in standard form , beginning with the nonzero term of highest degree and continuing with terms in descending order according to degree. If a power of x is missing, it is because its coefficient is zero. Polynomial functions are among the simplest in algebra. They are easy to evaluate: only addition and repeated multiplication are required. Because of this, they are often used to approximate other, more complicated functions. In this section, we investigate properties of this important class of functions.
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