3.3E: Graphs of Polynomial Functions (Exercises)

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Jan 2, 2021
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- 3.3: Graphs of Polynomial Functions
- 3.4: Factor Theorem and Remainder Theorem

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section 3.3 exercise

Find the $C$ and $t$ intercepts of each function.

1. $C\left(t\right)=2\left(t-4\right)\left(t+1\right)(t-6)$

2. $C\left(t\right)=3\left(t+2\right)\left(t-3\right)(t+5)$

3. $C\left(t\right)=4t\left(t-2\right)^{2} (t+1)$

4. $C\left(t\right)=2t\left(t-3\right)\left(t+1\right)^{2}$

5. $C\left(t\right)=2t^{4} -8t^{3} +6t^{2}$

6. $C\left(t\right)=4t^{4} +12t^{3} -40t^{2}$

Use your calculator or other graphing technology to solve graphically for the zeros of the function.

7. $f\left(x\right)=x^{3} -7x^{2} +4x+30$

8. $g\left(x\right)=x^{3} -6x^{2} +x+28$

Find the long run behavior of each function as $t \to \infty$ and $t \to -\infty$

9. $h\left(t\right)=3\left(t-5\right)^{3} \left(t-3\right)^{3} (t-2)$

10. $k\left(t\right)=2\left(t-3\right)^{2} \left(t+1\right)^{3} (t+2)$

11. $p\left(t\right)=-2t\left(t-1\right)\left(3-t\right)^{2}$

12. $q\left(t\right)=-4t\left(2-t\right)\left(t+1\right)^{3}$

Sketch a graph of each equation.

13. $f\left(x\right)=\left(x+3\right)^{2} (x-2)$

14. $g\left(x\right)=\left(x+4\right)\left(x-1\right)^{2}$

15. $h\left(x\right)=\left(x-1\right)^{3} \left(x+3\right)^{2}$

16. $k\left(x\right)=\left(x-3\right)^{3} \left(x-2\right)^{2}$

17. $m\left(x\right)=-2x\left(x-1\right)(x+3)$

18. $n\left(x\right)=-3x\left(x+2\right)(x-4)$

Solve each inequality.

19. $\left(x-3\right)\left(x-2\right)^{2} >0$

20. $\left(x-5\right)\left(x+1\right)^{2} >0$

21. $\left(x-1\right)\left(x+2\right)\left(x-3\right)<0$

22. $\left(x-4\right)\left(x+3\right)\left(x+6\right)<0$

Find the domain of each function.

23. $f\left(x\right)=\sqrt{-42+19x-2x^{2} }$

24. $g\left(x\right)=\sqrt{28-17x-3x^{2} }$

25. $h\left(x\right)=\sqrt{4-5x+x^{2} }$

26. $k\left(x\right)=\sqrt{2+7x+3x^{2} }$

27. $n\left(x\right)=\sqrt{\left(x-3\right)\left(x+2\right)^{2} }$

28. $m\left(x\right)=\sqrt{\left(x-1\right)^{2} (x+3)}$

29. $p\left(t\right)=\dfrac{1}{t^{2} +2t-8}$

30. $q\left(t\right)=\dfrac{4}{x^{2} -4x-5}$

Write an equation for a polynomial the given features.

31. Degree 3. Zeros at $x$ = -2, $x$ = 1, and $x$ = 3. Vertical intercept at (0, -4)

32. Degree 3. Zeros at $x$ = -5, $x$ = -2, and $x$ = 1. Vertical intercept at (0, 6)

33. Degree 5. Roots of multiplicity 2 at $x$ = 3 and $x$ = 1, and a root of multiplicity 1 at $x$ = -3. Vertical intercept at (0, 9)

34. Degree 4. Root of multiplicity 2 at $x$ = 4, and a roots of multiplicity 1 at $x$ = 1 and $x$ = -2. Vertical intercept at (0, -3)

35. Degree 5. Double zero at $x$ = 1, and triple zero at $x$ = 3. Passes through the point (2, 15)

36. Degree 5. Single zero at $x$ = -2 and $x$ = 3, and triple zero at $x$ = 1. Passes through the point (2, 4)

Write a formula for each polynomial function graphed.

37. $屏幕快照 2019-06-23 上午3.24.15.png$ 38. $屏幕快照 2019-06-23 上午3.24.35.png$ 39. $屏幕快照 2019-06-23 上午3.24.57.png$

40. $屏幕快照 2019-06-23 上午3.25.18.png$ 41. $屏幕快照 2019-06-23 上午3.25.44.png$ 42. $屏幕快照 2019-06-23 上午3.26.12.png$

43. $屏幕快照 2019-06-23 上午3.26.29.png$ 44. $屏幕快照 2019-06-23 上午3.26.46.png$

Write a formula for each polynomial function graphed.

45. $屏幕快照 2019-06-23 上午5.32.03.png$ 46. $屏幕快照 2019-06-23 上午5.32.31.png$

47. $屏幕快照 2019-06-23 上午5.32.47.png$ 48. $屏幕快照 2019-06-23 上午5.33.17.png$

49. $屏幕快照 2019-06-23 上午5.33.39.png$ 50. $屏幕快照 2019-06-23 上午5.34.00.png$

51. A rectangle is inscribed with its base on the $x$ axis and its upper corners on the parabola $y=5-x^{2}$ . What are the dimensions of such a rectangle that has the greatest possible area?

52. A rectangle is inscribed with its base on the $x$ axis and its upper corners on the curve $y=16-x^{4}$ . What are the dimensions of such a rectangle that has the greatest possible area?

Answer

$C(t)$	$C$ , intercepts	$t$ , intercepts
1.	(0, 48)	(4, 0), (-1, 0), (6, 0)
3.	(0, 0)	(0, 0), (2, 0), (-1, 0)
5.	(0, 0)	(0, 0), (1, 0), (3, 0)

7. (-1.646, 0) (3.646, 0) (5, 0)

9. As $t \to \infty$ , $h(t) \to \infty$ $t \to -\infty$ , $h(t) \to -\infty$

11. As $t \to \infty$ , $p(t) \to -\infty$ $t \to -\infty$ , $p(t) \to -\infty$

13. $Screen Shot 2019-10-03 at 6.05.35 PM.png$

15. $Screen Shot 2019-10-03 at 6.05.54 PM.png$

17. $Screen Shot 2019-10-03 at 6.06.33 PM.png$

19. $(3, \infty)$

21. $(-\infty, -2) \cup (1, 3)$

23. [3, 5, 6]

25. $(-\infty, 1] \cup [4, \infty)$

27. $[-2, -2] \cup [3, \infty)$

29. $(-\infty, -4) \cup (-4, 2) \cup (2, \infty)$

31. $y = -\dfrac{2}{3} (x + 2) (x - 1) (x - 3)$

33. $y = \dfrac{1}{3} (x - 1)^2 (x - 3)^2 (x + 3)$

35. $y = -15(x - 1)^2 (x - 3)^2$

37. $y = \dfrac{1}{2} (x + 2)(x - 1) (x - 3)$

39. $y = -(x + 1)^2 (x - 2)$

41. $y = -\dfrac{1}{24} (x + 3)(x + 2) (x - 2) (x - 4)$

43. $y = \dfrac{1}{24} (x + 4) (x + 2) (x - 3)^2$

45. $y = \dfrac{1}{12} (x + 2)^2 (x - 3)^2$

47. $y = \dfrac{1}{6} (x + 3) (x + 2) (x - 1)^3$

49. $y = -\dfrac{1}{16} (x + 3)(x + 1) (x - 2)^2 (x - 4)$

51. Base 2.58, Height 3.336

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