5.4.2: Homework

Reading Questions

1. What are two key visual characteristics of the graphs of polynomial functions of degree 2 or more?
2. What are the "zeros" of a polynomial function \(f(x)\), and how do they relate to the graph's \(x\)-intercepts?
3. Describe three methods mentioned for finding the zeros of a polynomial function.
4. What does the "multiplicity" of a zero tell us about the behavior of the graph of a polynomial function at that \(x\)-intercept?
   • What happens if the multiplicity is odd?
   • What happens if the multiplicity is even?
5. How does the graph's behavior near an \(x\)-intercept change for higher even or odd multiplicities?
6. What determines the end behavior of a polynomial function's graph?
7. What is the maximum number of turning points a polynomial function of degree \(n\) can have?
8. List the general steps suggested for sketching the graph of a polynomial function.
9. What does the Intermediate Value Theorem state about a polynomial function \(f\) if \(f(a)\) and \(f(b)\) have opposite signs?
10. If you know the \(x\)-intercepts and their multiplicities for a polynomial, how can you write a general formula for that polynomial function? What additional piece of information is needed to find the specific stretch factor \(a\)?
11. What is the difference between a local maximum/minimum and a global maximum/minimum for a polynomial function?
12. Do all polynomial functions have a global minimum or maximum? Explain.
13. How do you graph the function \(g(x) = |f(x)|\) if you know the graph of \(f(x)\), where \(f(x)\) is a polynomial function?
14. Describe a graphical approach to solving a polynomial inequality like \(P(x) > 0\).
15. What is the difference between an \( x \)-intercept and a zero of a polynomial function \(f\)?
16. If a polynomial function of degree \(n\) has \(n\) distinct zeros, what do you know about the graph of the function?
17. Explain how the Intermediate Value Theorem can assist us in finding a zero of a function.
18. Explain how the factored form of the polynomial helps us in graphing it.
19. If the graph of a polynomial just touches the \( x \)-axis and then changes direction, what can we conclude about the factored form of the polynomial?

Homework

Algebraic

For the following exercises, find the \( x \)- or \( t \)-intercepts of the polynomial functions.

20. \(C( t )=2( t−4 )( t+1 )(t−6)\)
21. \(C( t )=3( t+2 )( t−3 )(t+5)\)
22. \(C( t )=4t ( t−2 )^2 (t+1)\)
23. \(C( t )=2t( t−3 ) ( t+1 )^2\)
24. \(C( t )=2 t^4 −8 t^3 +6 t^2\)
25. \(C( t )=4 t^4 +12 t^3 −40 t^2\)
26. \(f(x)= x^4 − x^2\)
27. \(f(x)= x^3 + x^2 −20x\)
28. \(f(x)= x^3 +6 x^2 −7x\)
29. \(f(x)= x^3 + x^2 −4x−4\)
30. \(f(x)= x^3 +2 x^2 −9x−18\)
31. \(f(x)=2 x^3 − x^2 −8x+4\)
32. \(f(x)= x^6 −7 x^3 −8\)
33. \(f(x)=2 x^4 +6 x^2 −8\)
34. \(f(x)= x^3 −3 x^2 −x+3\)
35. \(f(x)= x^6 −2 x^4 −3 x^2\)
36. \(f(x)= x^6 −3 x^4 −4 x^2\)
37. \(f(x)= x^5 −5 x^3 +4x\)

For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval.

38. \(f(x)= x^3 −9x\), between \(x=−4\) and \(x=−2\).
39. \(f(x)= x^3 −9x\), between \(x=2\) and \(x=4\).
40. \(f(x)= x^5 −2x\), between \(x=1\) and \(x=2\).
41. \(f(x)=− x^4 +4\), between \(x=1\) and \(x=3\) .
42. \(f(x)=−2 x^3 −x\), between \(x=–1\) and \(x=1\).
43. \(f(x)= x^3 −100x+2\), between \(x=0.01\) and \(x=0.1\)

For the following exercises, find the zeros and give the multiplicity of each.

44. \(f(x)= ( x+2 )^3 ( x−3 )^2\)
45. \(f(x)= x^2 ( 2x+3 )^5 ( x−4 )^2\)
46. \(f(x)= x^3 ( x−1 )^3 ( x+2 )\)
47. \(f(x)= x^2 ( x^2 +4x+4 )\)
48. \(f(x)= ( 2x+1 )^3 ( 9 x^2 −6x+1 )\)
49. \(f(x)= ( 3x+2 )^5 ( x^2 −10x+25 )\)
50. \(f(x)=x( 4 x^2 −12x+9 )( x^2 +8x+16 )\)
51. \(f(x)= x^6 − x^5 −2 x^4\)
52. \(f(x)=3 x^4 +6 x^3 +3 x^2\)
53. \(f(x)=4 x^5 −12 x^4 +9 x^3\)
54. \(f(x)=2 x^4 ( x^3 −4 x^2 +4x )\)
55. \(f(x)=4 x^4 ( 9 x^4 −12 x^3 +4 x^2 )\)

Graphical

For the following exercises, graph the polynomial functions. Note \( x \)- and \( y \)-intercepts, multiplicity, and end behavior.

56. \(f( x )= ( x+3 )^2 (x−2)\)
57. \(g( x )=( x+4 ) ( x−1 )^2\)
58. \(h( x )= ( x−1 )^3 ( x+3 )^2\)
59. \(k( x )= ( x−3 )^3 ( x−2 )^2\)
60. \(m( x )=−2x( x−1 )(x+3)\)
61. \(n( x )=−3x( x+2 )(x−4)\)

For the following exercises, use the graph to identify zeros and multiplicity.

62.  

    

63.  

    

64. 
    

65. 
    

Technology

For the following exercises, use a calculator to approximate local minima and maxima or the global minimum and maximum.

66. \(f(x)= x^3 −x−1\)
67. \(f(x)=2 x^3 −3x−1\)
68. \(f(x)= x^4 +x\)
69. \(f(x)=− x^4 +3x−2\)
70. \(f(x)= x^4 − x^3 +1\)

Real-World Applications

For the following exercises, write the polynomial function that models the given situation.

71. A rectangle has a length of 10 units and a width of 8 units. Squares of \(x\) by \(x\) units are cut out of each corner, and then the sides are folded up to create an open box. Express the volume of the box as a polynomial function in terms of \(x\).
72. Consider the same rectangle of the preceding problem. Squares of \(2x\) by \(2x\) units are cut out of each corner. Express the volume of the box as a polynomial in terms of \(x\).
73. A square has sides of 12 units. Squares \(x+1\) by \(x+1\) units are cut out of each corner, and then the sides are folded up to create an open box. Express the volume of the box as a function in terms of \(x\).
74. A cylinder has a radius of \(x+2\) units and a height of 3 units greater. Express the volume of the cylinder as a polynomial function.
75. A right circular cone has a radius of \(3x+6\) and a height 3 units less. Express the volume of the cone as a polynomial function. The volume of a cone is \(V= \frac{1}{3} \pi r^2 h\) for radius \(r\) and height \( h\).