Chap6 Section2

Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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Section 6.2 Zeros of Polynomials 577 Version: Fall 2007 6.2 Exercises In Exercises 1 - 6 , use direct substitu- tion to show that the given value is a zero of the given polynomial. 1. p ( x ) = x 3 3 x 2 13 x + 15 , x = 3 2. p ( x ) = x 3 2 x 2 13 x 10 , x = 2 3. p ( x ) = x 4 x 3 12 x 2 , x = 4 4. p ( x ) = x 4 2 x 3 3 x 2 , x = 1 5. p ( x ) = x 4 + x 2 20 , x = 2 6. p ( x ) = x 4 + x 3 19 x 2 + 11 x + 30 , x = 1 In Exercises 7 - 28 , identify all of the zeros of the given polynomial without the aid of a calculator. Use an alge- braic technique and show all work (fac- tor when necessary) needed to obtain the zeros. 7. p ( x ) = ( x 2)( x + 4)( x 5) 8. p ( x ) = ( x 1)( x 3)( x + 8) 9. p ( x ) = 2( x 3)( x + 4)( x 2) 10. p ( x ) = 3( x + 1)( x 1)( x 8) 11. p ( x ) = x ( x 3)(2 x + 1) 12. p ( x ) = 3 x ( x + 5)(3 x 2) 13. p ( x ) = 2( x + 3)(3 x 5)(2 x + 1) 14. p ( x ) = 3( x 2)(2 x + 5)(3 x 4) 15. p ( x ) = 3 x 3 + 5 x 2 12 x 20 Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/ 1 16. p ( x ) = 3 x 3 + x 2 12 x 4 17. p ( x ) = 2 x 3 + 5 x 2 2 x 5 18. p ( x ) = 2 x 3 5 x 2 18 x + 45 19. p ( x ) = x 4 + 4 x 3 9 x 2 36 x 20. p ( x ) = x 4 + 4 x 3 + x 2 4 x 21. p ( x ) = 2 x 4 10 x 3 + 8 x 2 + 40 x 22. p ( x ) = 3 x 4 + 6 x 3 75 x 2 150 x 23. p ( x ) = 2 x 3 7 x 2 15 x 24. p ( x ) = 2 x 3 x 2 10 x 25. p ( x ) = 6 x 3 + 4 x 2 + 16 x 26. p ( x ) = 9 x 3 + 3 x 2 30 x 27. p ( x ) = 2 x 7 10 x 6 + 8 x 5 + 40 x 4 28. p ( x ) = 6 x 5 21 x 4 45 x 3
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578 Chapter 6 Polynomial Functions Version: Fall 2007 In Exercises 29 - 34 , the graph of a poly- nomial is given. Perform each of the fol- lowing tasks. i. Copy the image onto your homework paper. Label and scale your axes, then label each x -intercept with its coordinates. ii. Identify the zeros of the polynomial. 29. x 10 y 10 30. x 10 y 10 31. x 10 y 10 32. x 10 y 10 33. x 10 y 10 34. x 10 y 10
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Section 6.2 Zeros of Polynomials 579 Version: Fall 2007 For each of the polynomials in Exercises 35 - 46 , perform each of the following tasks. i. Factor the polynomial to obtain the zeros. Show your work. ii. Set up a coordinate system on graph paper. Label and scale the horizontal axis. Use the zeros and end-behavior to help sketch the graph of the poly- nomial without the use of a calcula- tor. iii. Verify your result with a graphing cal- culator. 35. p ( x ) = 5 x 3 + x 2 45 x 9 36. p ( x ) = 4 x 3 + 3 x 2 64 x 48 37. p ( x ) = 4 x 3 12 x 2 9 x + 27 38. p ( x ) = x 3 + x 2 16 x 16 39. p ( x ) = x 4 + 2 x 3 25 x 2 50 x 40. p ( x ) = x 4 5 x 3 + 4 x 2 + 20 x 41. p ( x ) = 3 x 4 9 x 3 + 3 x 2 + 9 x 42. p ( x ) = 4 x 4 29 x 2 + 25 43. p ( x ) = x 3 x 2 + 20 x 44. p ( x ) = 2 x 3 7 x 2 30 x 45. p ( x ) = 2 x 3 + 3 x 2 35 x 46. p ( x ) = 2 x 3 11 x 2 + 21 x
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Chapter 6 Polynomial Functions Version: Fall 2007 6.2 Solutions 1. p ( 3) = ( 3) 3 3( 3) 2 13( 3) + 15 = 27 27 + 39 + 15 = 0 3. p (4) = 4 4 4 3 12(4) 2 = 256 64 192 = 0 5. p ( 2) = ( 2) 4 + ( 2) 2 20 = 16 + 4 20 = 0 7. Set p ( x ) = 0 in p ( x ) = ( x 2)( x + 4)( x 5) , 0 = ( x 2)( x + 4)( x 5) , then use the zero product property to write x 2 = 0 or x + 4 = 0 or x 5 = 0 .
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