Chapter Review 267 Section You should be able to . . . Examples Review Exercises 4.2 1 Analyze the graph of a polynomial function (p. 206) 1–3 8–11 2 Build cubic models from data (p. 210) 4 51 4.3 1 Use the Remainder and Factor Theorems (p. 215) 1, 2 12–14 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 15, 16 3 Use the Rational Zeros Theorem to list the potential rational zeros of a polynomial function (p. 218) 4 17–20 4 Find the real zeros of a polynomial function (p. 219) 5, 6 18–20 5 Solve polynomial equations (p. 222) 7 21, 22 6 Use the Theorem for Bounds on Zeros (p. 222) 8–10 23, 24 7 Use the Intermediate Value Theorem (p. 225) 11 25–28 4.4 1 Use the Conjugate Pairs Theorem (p. 231) 1 29, 30 2 Find a polynomial function with specified zeros (p. 232) 2 29, 30 3 Find the complex zeros of a polynomial function (p. 233) 3 31–34 4.5 1 Find the domain of a rational function (p. 236) 1–3 35, 36 2 Find the vertical asymptotes of a rational function (p. 240) 4 35, 36, 44 3 Find a horizontal or an oblique asymptote of a rational function (p. 241) 5–8 35, 36, 44 4.6 1 Analyze the graph of a rational function (p. 247) 1–6 37–42 2 Solve applied problems involving rational functions (p. 254) 7 50 4.7 1 Solve polynomial inequalities graphically and algebraically (p. 259) 1, 2 43, 45, 46 2 Solve rational inequalities graphically and algebraically (p. 261) 3, 4 44, 47–49 Review Exercises In Problems 1–4, determine whether the function is a polynomial function, rational function, or neither. For those that are polynomial functions, state the degree. For those that are not polynomial functions, tell why not. 1. ( ) = − + − f x x x x 4 3 5 2 5 2 2. ( ) = + f x x x 3 2 1 5 3. ( ) = + − f x x x 3 5 1 2 1/2 4. ( ) = f x 3 In Problems 5–7, graph each function using transformations (shifting, compressing, stretching, and reflection). Show all the steps. 5. ( ) ( ) = + f x x 2 3 6. ( ) ( ) = − − f x x 1 4 7. ( ) ( ) = − + f x x 1 2 4 In Problems 8–11, analyze each polynomial function by following Steps 1 through 8 on page 207. 8. ( ) ( )( ) = + + f x x x x 2 4 9. ( ) ( ) ( ) = − + f x x x 2 4 2 10. ( ) = − + f x x x 2 4 3 2 11. ( ) ( ) ( )( ) = − + + f x x x x 1 3 1 2 In Problems 12 and 13, find the remainder R when ( ) f x is divided by ( ) g x . Is g a factor of f? 12. ( ) ( ) = − + + = − f x x x x g x x 8 3 4; 1 3 2 13. ( ) ( ) = − + − = + f x x x x g x x 2 15 2; 2 4 3 14. Find the value of ( ) = − + f x x x 12 8 1 6 4 at = x 4. In Problems 15 and 16, use Descartes’ Rule of Signs to determine how many positive and how many negative zeros each polynomial function may have. Do not attempt to find the zeros. 15. ( ) = − + − + + f x x x x x x 12 8 2 3 8 7 4 3 16. ( ) = − + + + + f x x x x x 6 5 1 5 4 3 17. List all the potential rational zeros of ( ) = − + − + − f x x x x x x 12 6 3 8 7 4 3 . In Problems 18–20, use the Rational Zeros Theorem to find all the real zeros of each polynomial function. Use the zeros to factor f over the real numbers. 18. ( ) = − − + f x x x x 3 6 8 3 2 19. ( ) = + − + f x x x x 4 4 7 2 3 2 20. ( ) = − + − + f x x x x x 4 9 20 20 4 3 2 In Problems 21 and 22, solve each equation in the real number system. 21. + − + − = x x x x 2 2 11 6 0 4 3 2 22. + + − − = x x x x 2 7 7 3 0 4 3 2 In Problems 23 and 24, find bounds to the real zeros of each polynomial function. Obtain a complete graph of f using a graphing utility. 23. ( ) = − − + f x x x x4 2 3 2 24. ( ) = − − + f x x x x 2 7 10 35 3 2 In Problems 25 and 26, use the Intermediate Value Theorem to show that each polynomial function has a zero in the given interval. 25. ( ) [ ] = − − f x x x 3 1; 0, 1 3 26. ( ) [ ] = − − − f x x x x 8 4 2 1; 0, 1 4 3
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