268 CHAPTER 4 Polynomial and Rational Functions In Problems 27 and 28, each polynomial function has exactly one positive zero. Approximate the zero correct to two decimal places. 27. ( ) = − − f x x x 2 3 28. ( ) = − − − f x x x x 8 4 2 1 4 3 In Problems 29 and 30, information is given about a complex polynomial function ( ) f x whose coefficients are real numbers. Find the remaining zeros of f . Then find a polynomial function with real coefficients that has the zeros. 29. Degree 3; zeros: +i 4 , 6 30. Degree 4; zeros: i, +i 1 In Problems 31–34, find the complex zeros of each polynomial function ( ) f x . Write f in factored form. 31. ( ) = − − + f x x x x 3 6 8 3 2 32. ( ) = + − + f x x x x 4 4 7 2 3 2 33. ( ) = − + − + f x x x x x 4 9 20 20 4 3 2 34. ( ) = + − + − f x x x x x 2 2 11 6 4 3 2 In Problems 35 and 36, find the domain of each rational function. Find any horizontal, vertical, or oblique asymptotes. 35. ( ) = + − R x x x 2 9 2 36. ( ) ( ) = + + + R x x x x 3 2 2 2 2 In Problems 37–42, analyze each rational function following Steps 1–7 given on page 249. 37. ( ) = − R x x x 2 6 38. ( ) ( ) = + − H x x x x 2 2 39. ( ) = + − − − R x x x x x 6 6 2 2 40. ( ) = − F x x x 4 3 2 41. ( ) ( ) = − R x x x 2 1 4 2 42. ( ) = − − − G x x x x 4 2 2 2 43. Use the graph below of a polynomial function ( ) = y f x to solve (a) ( ) = f x 0, (b) ( ) > f x 0, (c) ( ) ≤ f x 0, and (d) determine f. x 4 2 24 22 y 18 14 6 10 2 22 y 5 f(x) 44. Use the graph below of a rational function ( ) = y R x to (a) identify the horizontal asymptote of R, (b) identify the vertical asymptotes of R, (c) solve ( ) < R x 0, (d) solve ( ) ≥ R x 0, and (e) determine R. x 4 2 24 22 y 2 4 24 22 x 5 22 x 5 2 y 5 0.25 0, ( ) 2 3 – 16 (23, 0) (21, 0) y 5 R(x) In Problems 45–49, solve each inequality. Graph the solution set. 45. + < + x x x4 4 3 2 46. + ≥ + x x x 4 4 3 2 47. − − < x x 2 6 1 2 48. ( )( ) − − − ≥ x x x 2 1 3 0 49. − + − > x x x 8 12 16 0 2 2 50. Making a Can A can in the shape of a right circular cylinder is required to have a volume of 250 cubic centimeters. (a) Express the amount A of material to make the can as a function of the radius r of the cylinder. (b) How much material is required if the can is of radius 3 centimeters? (c) How much material is required if the can is of radius 5 centimeters? (d) Graph ( ) = A A r . For what value of r is A smallest? 51. Housing Prices The data in the table represent the January median new-home prices in the United States for the years shown. (a) With a graphing utility, draw a scatter diagram of the data. Comment on the type of relation that appears to exist between the two variables. (b) Decide on the function of best fit to these data (linear, quadratic, or cubic), and use this function to predict the median new-home price in the United States for January 2023 ( ) = t 8 . (c) Draw the function of best fit on the scatter diagram obtained in part (a). Year, t Median Price, ( ) P $1000s 2016, 1 299.8 2017, 2 313.1 2018, 3 331.8 2019, 4 313.0 2020, 5 329.0 2021, 6 369.8 2022, 7 433.1 Source: Census, Housing & Urban Development.
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