Chapter 6: Exercises with Solutions

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

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Section 6.1 Polynomial Functions 559 Version: Fall 2007 6.1 Exercises In Exercises 1 - 8 , arrange each polyno- mial in descending powers of x , state the degree of the polynomial, identify the lead- ing term, then make a statement about the coe ffi cients of the given polynomial. 1. p ( x ) = 3 x - x 2 + 4 - x 3 2. p ( x ) = 4 + 3 x 2 - 5 x + x 3 3. p ( x ) = 3 x 2 + x 4 - x - 4 4. p ( x ) = - 3 + x 2 - x 3 + 5 x 4 5. p ( x ) = 5 x - 3 2 x 3 + 4 - 2 3 x 5 6. p ( x ) = - 3 2 x + 5 - 7 3 x 5 + 4 3 x 3 7. p ( x ) = - x + 2 3 x 3 - 2 x 2 + π x 6 8. p ( x ) = 3+ 2 x 4 + 3 x - 2 x 2 + 5 x 6 In Exercises 9 - 14 , you are presented with the graph of y = ax n . In each case, state whether the degree is even or odd, then state whether a is a positive or negative number. 9. x 5 y 5 Copyrighted material. See: 1 10. x 5 y 5 11. x 5 y 5 12. x 5 y 5

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560 Chapter 6 Polynomial Functions Version: Fall 2007 13. x 5 y 5 14. x 5 y 5 In Exercises 15 - 20 , you are presented with the graph of the polynomial p ( x ) = a n x n + · · · + a 1 x + a 0 . In each case, state whether the degree of the polynomial is even or odd, then state whether the lead- ing coe ffi cient a n is positive or negative. 15. x y 16. x y 17. x y 18. x y
Section 6.1 Polynomial Functions 561 Version: Fall 2007 19. x y 20. x y For each polynomial in Exercises 21 - 30 , perform each of the following tasks. i. Predict the end-behavior of the poly- nomial by drawing a very rough sketch of the polynomial. Do this without the assistance of a calculator. The only concern here is that your graph show the correct end-behavior. ii. Draw the graph on your calculator, adjust the viewing window so that all “turning points” of the polyno- mial are visible in the viewing win- dow, and copy the result onto your homework paper. As usual, label and scale each axis with xmin, xmax, ymin, and ymax. Does the actual end-behavior agree with your predicted end-behavior? 21. p ( x ) = - 3 x 3 + 2 x 2 + 8 x - 4 22. p ( x ) = 2 x 3 - 3 x 2 + 4 x - 8 23. p ( x ) = x 3 + x 2 - 17 x + 15 24. p ( x ) = - x 4 + 2 x 2 + 29 x - 30 25. p ( x ) = x 4 - 3 x 2 + 4 26. p ( x ) = - x 4 + 8 x 2 - 12 27. p ( x ) = - x 5 + 3 x 4 - x 3 + 2 x 28. p ( x ) = 2 x 4 - 3 x 3 + x - 10 29. p ( x ) = - x 6 - 4 x 5 + 27 x 4 + 78 x 3 + 4 x 2 + 376 x - 480 30. p ( x ) = x 5 - 27 x 3 +30 x 2 - 124 x +120

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Chapter 6 Polynomial Functions Version: Fall 2007 6.1 Solutions 1. p ( x ) = - x 3 - x 2 +3 x +4 , degree = 3 , leading term = - x 3 , “ p is a polynomial with integer coe ffi cients,” “ p is a polynomial with rational coe ffi cients,” or “ p is a polynomial with real coe ffi cients.” 3. p ( x ) = x 4 + 3 x 2 - x - 4 , degree = 4 , leading term = x 4 , “ p is a polynomial with integer coe ffi cients,” “ p is a polynomial with rational coe ffi cients,” or “ p is a polynomial with real coe ffi cients.” 5. p ( x ) = - 2 3 x 5 - 3 2 x 3 + 5 x + 4 , degree = 5 , leading term = - 2 3 x 5 , “ p is a polynomial with rational coe ffi cients,” or “ p is a polynomial with real coe ffi cients.” 7. p ( x ) = π x 6 + 2 3 x 3 - 2 x 2 - x , degree = 6 , leading term = π x 6 , “ p is a polynomial with real coe ffi cients.” 9. Note that the graph of the given function, f ( x ) = ax n , has di ff erent end-behavior at its left- and right-ends.
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