214 CHAPTER 4 Polynomial and Rational Functions ‘Are You Prepared?’ Answers 1. ( ) ( ) − 0, 10 , 2, 0 2. x7 3 − 3. Local maximum value 6.48 at x 0.67; = − local minimum value 3− at x 2 = 4. y x x 0.337 2.311 0.216 2 = − + − Retain Your Knowledge Problems 50–59 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for subsequent sections, a final exam, or later courses such as calculus. 50. Solve x 2 3 1 4 10. − + > 51. Determine the function that is graphed if the graph of f x x ( ) = is reflected about the x -axis and then vertically compressed by a factor of 1 3 . 52. Find the vertex of the graph of f x x x 2 7 3. 2 ( ) = − + − 53. The strain on a solid object varies directly with the external tension force acting on the solid and inversely with the cross-sectional area. If a certain steel rod has a strain of 150 when the force is 1.47 10 N 4 × and the cross-sectional area is 4.9 10 m , 4 2 × − find the strain for a similar rod with a cross-sectional area of 8.75 10 m3 2 × − and a tension force of 2.45 10 N. 5 × 54. Given f x x x 2 7 1, 3 ( ) = − + find f 1 2 . ( ) − 55. Find the domain of f x x 9 4 1. ( ) = − − + 56. Find the average rate of change of f x x x4 3 2 ( ) = + − from 2− to 1. 57. Find the center and radius of the circle x x y y 4 2 11 2 2 + + − = 58. Determine if the function g x x x x 3 3 ( ) = − is even, odd, or neither. 59. How long will it take $5000 to grow to $7500 at a simple interest rate of 8%? OBJECTIVES 1 Use the Remainder and Factor Theorems (p. 215) 2 Use Descartes’ Rule of Signs to Determine the Number of Positive and the Number of Negative Real Zeros of a Polynomial Function (p. 217) 3 Use the Rational Zeros Theorem to List the Potential Rational Zeros of a Polynomial Function (p. 218) 4 Find the Real Zeros of a Polynomial Function (p. 219) 5 Solve Polynomial Equations (p. 222) 6 Use the Theorem for Bounds on Zeros (p. 222) 7 Use the Intermediate Value Theorem (p. 225) 4.3 The Real Zeros of a Polynomial Function Now Work the ‘Are You Prepared?’ problems on page 226. • Evaluating Functions (Section 2.1, pp. 65–67) • Factoring Polynomials (Section A.3, pp. A27–A28) • Synthetic Division (Section A.4, pp. A31–A34) • Polynomial Division (Section A.3, pp. A25–A27) • Solving Quadratic Equations (Section A.6, pp. A48–A54) PREPARING FOR THIS SECTION Before getting started, review the following: In Section 4.1, we were able to identify the real zeros of a polynomial function because either the polynomial function was in factored form or it could be easily factored. But how do we find the real zeros of a polynomial function if it is not factored or cannot be easily factored? Recall that if r is a real zero of a polynomial function f , then f r 0, ( ) = r is an x -intercept of the graph of f , x r − is a factor of f , and r is a solution of the equation f x 0 ( ) = . For example, if x 4 − is a factor of f , then 4 is a real zero of f , and 4 is a solution to the equation f x 0 ( ) = . For polynomial functions, we have seen the importance of the real zeros for graphing. In most cases, however, the real zeros of a polynomial function are difficult to find using algebraic methods. No nice formulas like the quadratic formula are available to help us find zeros for polynomials of degree 3 or higher. Formulas do exist for solving any third- or fourth-degree polynomial equation, but they are somewhat complicated. No general formulas exist for polynomial equations of degree 5 or higher. Refer to the Historical Feature at the end of this section for more information.
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