196 CHAPTER 4 Polynomial and Rational Functions Seeing the Concept Graph the function found in Example 4 for a 2 = and a 1. =− Does the value of a affect the real zeros of f ? How does the value of a affect the graph of f ? Now Work PROBLEM 43 If, when f is factored, the factor x r − occurs more than once, r is called a repeated , or multiple , real zero of f. More precisely, we have the following definition. DEFINITION Real Zero of Multiplicity m If x r m ( ) − is a factor of a polynomial f and x r m 1 ( ) − + is not a factor of f, then r is called a real zero of multiplicity m of f. * Now Work PROBLEM 61(a) In Example 5, notice that if you add the multiplicities 2 1 4 7 , ( ) + + = you obtain the degree of the polynomial function. Suppose that it is possible to factor completely a polynomial function and, as a result, locate all the x -intercepts of its graph (the real zeros of the function). The following example illustrates the role that the multiplicity of an x -intercept plays. Identifying Real Zeros and Their Multiplicities For the polynomial f x x x x 5 2 1 2 2 4 ( ) ( ) ( ) = + − • 0 is a real zero of multiplicity 2 because the exponent on the factor x is 2. • 2− is a real zero of multiplicity 1 because the exponent on the factor x 2 + is 1. • 1 2 is a real zero of multiplicity 4 because the exponent on the factor x 1 2 − is 4. EXAMPLE 5 In Words The multiplicity of a real zero is the number of times its corresponding factor occurs. Figure 8 f x x x 1 2 2 ( ) ( ) ( ) = + − Investigating the Role of Multiplicity For the polynomial function f x x x 1 2 : 2 ( ) ( ) ( ) = + − (a) Find the x - and y -intercepts of the graph of f. (b) Using a graphing utility, graph the polynomial function. (c) For each x -intercept, determine whether it is of odd or even multiplicity. EXAMPLE 6 Solution (a) The y- intercept is f 0 0 1 0 2 2. 2 ( ) ( ) ( ) = + − = − The x -intercepts satisfy the equation f x x x 1 2 0 2 ( ) ( ) ( ) = + − = from which we find that ( ) + = =− − = = x x x x 1 0 1 or or 2 0 2 2 The x -intercepts are 1− and 2. (b) See Figure 8 for the graph of f using Desmos. (c) We can see from the factored form of f that 1− is a zero or root of multiplicity 2, and 2 is a zero or root of multiplicity 1; so 1− is of even multiplicity and 2 is of odd multiplicity. *Some texts use the terms multiple root and root of multiplicity m.
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