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Mathematics LibreTexts

3.3E: Exercises

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    17863
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    Section Exercises

    Verbal

    1. Explain the difference between the coefficient of a power function and its degree.

    Answer: The coefficient of the power function is the real number that is multiplied by the variable raised to a power. The degree is the highest power appearing in the function.

    2. If a polynomial function is in factored form, what would be a good first step in order to determine the degree of the function?

    3. In general, explain the end behavior of a power function with odd degree if the leading coefficient is positive.

    Answer: As \(x\) decreases without bound, so does \(f(x)\). As \(x\) increases without bound, so does \(f(x)\).

    4. What is the relationship between the degree of a polynomial function and the maximum number of turning points in its graph?

    5. What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As x→−∞, \(f(x)→−∞\) and as x→∞, \(f(x)→−∞\).

    Answer: The polynomial function is of even degree and leading coefficient is negative.

    Algebraic

    For the following exercises, identify the function as a power function, a polynomial function, or neither.


    6. \(f(x)=x^5\)

    7. \(f(x)=(x^2)^3\)

    Answer: Power function

    8. \(f(x)=x−x^4\)

    Exercise \(\PageIndex{9}\)

    \(f(x)=\frac{x^2}{x^2−1}\)

    Answer

    Neither

    10. \(f(x)=2x(x+2)(x−1)^2\)

    11. \(f(x)=3^{x+1}\)

    Answer: Neither

    For the following exercises, find the degree and leading coefficient for the given polynomial.


    12. \(−3x^4\)

    13. \(7−2x^2\)

    Answer: Degree = 2, Coefficient = –2

    14. \(−2x^2− 3x^5+ x−6\)

    15. \(x(4−x^2)(2x+1)\)

    Answer: Degree =4, Coefficient = –2

    16. \(x^2(2x−3)^2\)

    For the following exercises, determine the end behavior of the functions.


    17. \(f(x)=x^4\)

    Answer: As \(x→∞\), \(f(x)→∞\), as \(x→−∞\), \(f(x)→∞\)

    18. \(f(x)=x^3\)

    19. \(f(x)=−x^4\)

    Answer: As \(x→−∞\), \(f(x)→−∞\), as \(x→∞\), \(f(x)→−∞\)

    20. \(f(x)=−x^9\)

    21. \(f(x)=−2x^4− 3x^2+ x−1\)

    Answer: As \(x→−∞\), \(f(x)→−∞\), as \(x→∞\), \(f(x)→−∞\)

    22. \(f(x)=3x^2+ x−2\)

    23. \(f(x)=x^2(2x^3−x+1)\)

    Answer: As \(x→∞\), \(f(x)→∞\), as \(x→−∞\), \(f(x)→−∞\)

    24. \(f(x)=(2−x)^7\)

    For the following exercises, find the intercepts of the functions.


    25. \(f(t)=2(t−1)(t+2)(t−3)\)

    Answer: y-intercept is \((0,12)\), t-intercepts are \((1,0);(–2,0); \) and \((3,0)\).

    26. \(g(n)=−2(3n−1)(2n+1)\)

    Exercise \(\PageIndex{27}\)

    \(f(x)=x^4−16\)

    Answer

    y-intercept is \((0,−16).\) x-intercepts are \((2,0)\) and \((−2,0)\).

    28. \(f(x)=x^3+27\)

    29. \(f(x)=x(x^2−2x−8)\)

    Answer: y-intercept is \((0,0)\). x-intercepts are \((0,0),(4,0),\) and \((−2, 0)\).

    30. \(f(x)=(x+3)(4x^2−1)\)

    Graphical

    For the following exercises, determine the least possible degree of the polynomial function shown.

    31.

    CNX_Precalc_Figure_03_03_201.jpg

    Answer: 3

    32.

    CNX_Precalc_Figure_03_03_202.jpg

    33.
    CNX_Precalc_Figure_03_03_203.jpg

    Answer: 5

    34.

    CNX_Precalc_Figure_03_03_204.jpg

    35.
    CNX_Precalc_Figure_03_03_205.jpg

    Answer: 3

    36.

    CNX_Precalc_Figure_03_03_206.jpg

    37.
    CNX_Precalc_Figure_03_03_207.jpg

    Answer: 5

    38.

    CNX_Precalc_Figure_03_03_208.jpg

    For the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If so, determine the number of turning points and the least possible degree for the function.

    39.

    CNX_Precalc_Figure_03_03_209.jpg

    Answer: Yes. Number of turning points is 2. Least possible degree is 3.

    40.

    CNX_Precalc_Figure_03_03_210.jpg

    41.
    CNX_Precalc_Figure_03_03_211.jpg

    Answer: Yes. Number of turning points is 1. Least possible degree is 2.

    42.

    CNX_Precalc_Figure_03_03_212.jpg

    43.
    CNX_Precalc_Figure_03_03_213.jpg

    Answer: Yes. Number of turning points is 0. Least possible degree is 3.

    Exercise \(\PageIndex{44}\)

    CNX_Precalc_Figure_03_03_214.jpg

    Answer

    No (the graph is not smooth)

    45.

    CNX_Precalc_Figure_03_03_215.jpg

    Answer: Yes. Number of turning points is 0. Least possible degree is 1.

    Numeric

    For the following exercises, make a table to confirm the end behavior of the function.


    46. \(f(x)=−x^3\)

    47. \(f(x)=x^4−5x^2\)

    Answer:

    \(x\) \(f(x)\)
    10 9,500
    100 99,950,000
    –10 9,500
    –100 99,950,000

    as \(x→−∞,\) \(f(x)→∞\), as \(x→∞,\) \(f(x)→∞\)

    48. \(f(x)=x^2(1−x)^2\)

    49. \(f(x)=(x−1)(x−2)(3−x)\)

    Answer:

    \(x\) \(f(x)\)
    10 9,500
    100 99,950,000
    –10 9,500
    –100 99,950,000

    as \(x→−∞,\) \(f(x)→∞\), as \(x→∞,\) \(f(x)→−∞\)

    50. \(f(x)=\frac{x^5}{10}−x^4\)

    Technology

    For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.

    51. \(f(x)=x^3(x−2)\)

    Answer:

    CNX_Precalc_Figure_03_03_216.jpg

    The y-intercept is \((0, 0)\). The x-intercepts are \((0, 0), (2, 0).\) As \(x→−∞,\) \(f(x)→∞\), as \(x→∞,\) \(f(x)→∞\)

    52. \(f(x)=x(x−3)(x+3)\)

    53. \(f(x)=x(14−2x)(10−2x)\)

    Answer:

    CNX_Precalc_Figure_03_03_218.jpg

    The y-intercept is \((0,0)\) . The x-intercepts are \((0, 0), (5, 0), (7, 0)\). As \(x→−∞\), \(f(x)→−∞\), as \(x→∞,\) \(f(x)→∞\)

    54. \(f(x)=x(14−2x)(10−2x)^2\)

    55. \(f(x)=x^3−16x\)

    Answer:

    CNX_Precalc_Figure_03_03_220.jpg

    The y-intercept is (0, 0). The x-intercept is \((−4, 0), (0, 0), (4, 0)\). As \(x→−∞\), \(f(x)→−∞\), as \(x→∞,\) \(f(x)→∞\)

    56. \(f(x)=x^3−27\)

    57. \(f(x)=x^4−81\)

    Answer:

    CNX_Precalc_Figure_03_03_222.jpg

    The y-intercept is (0, −81). The x-intercept are \((3, 0), (−3, 0)\). As \(x→−∞,\) \(f(x)→∞\), as \(x→∞,\) \(f(x)→∞\)

    58. \(f(x)=−x^3+x^2+2x\)

    59. \(f(x)=x^3−2x^2−15x\)

    Answer:

    CNX_Precalc_Figure_03_03_224.jpg

    The y-intercept is \((0, 0)\). The x-intercepts are \((−3, 0), (0, 0), (5, 0).\) As \(x→−∞\), \(f(x)→−∞\), as \(x→∞,\) \(f(x)→∞\)

    60. \(f(x)=x^3−0.01x\)

    Extensions

    For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or –1. There may be more than one correct answer.

    61. The y-intercept is \((0,−4)\). The x-intercepts are \((−2,0), (2,0)\). Degree is 2.

    End behavior: as \(x→−∞,\) \(f(x)→∞\), as \(x→∞,\) \(f(x)→∞\).

    Answer: \(f(x)=x^2−4\)

    62. The y-intercept is \((0,9)\). The x-intercepts are \((−3,0), (3,0)\). Degree is 2.

    End behavior: as \(x→−∞,\) \(f(x)→−∞\), as \(x→∞,\) \(f(x)→−∞\).

    63. The y-intercept is \((0,0)\). The x-intercepts are \((0,0), (2,0)\). Degree is 3.

    End behavior: as \(x→−∞,\) \(f(x)→−∞\), as \(x→∞,\) \(f(x)→∞\).

    Answer: \(f(x)=x^3−4x^2+4x\)

    64. The y-intercept is \((0,1)\). The x-intercept is \((1,0)\). Degree is 3.

    End behavior: as \(x→−∞\), \(f(x)→∞\), as \(x→∞\), \(f(x)→−∞\).

    65. The y-intercept is \((0,1)\). There is no x-intercept. Degree is 4.

    End behavior: as \(x→−∞,\) \(f(x)→∞\), as \(x→∞,\) \(f(x)→∞\).

    Answer: \(f(x)=x^4+1\)

    Real-World Applications

    For the following exercises, use the written statements to construct a polynomial function that represents the required information.

    66. An oil slick is expanding as a circle. The radius of the circle is increasing at the rate of 20 meters per day. Express the area of the circle as a function of \(d\), the number of days elapsed.

    67. A cube has an edge of 3 feet. The edge is increasing at the rate of 2 feet per minute. Express the volume of the cube as a function of \(m\), the number of minutes elapsed.

    Answer: \(V(m)=8m^3+36m^2+54m+27\)

    68. A rectangle has a length of 10 inches and a width of 6 inches. If the length is increased by \(x\) inches and the width increased by twice that amount, express the area of the rectangle as a function of \(x\).

    Exercise \(\PageIndex{69}\)

    An open box is to be constructed by cutting out square corners of \(x\)-inch sides from a piece of cardboard 8 inches by 8 inches and then folding up the sides. Express the volume of the box as a function of \(x\).

    Answer

    \(V(x)=4x^3−32x^2+64x\)

    70. A rectangle is twice as long as it is wide. Squares of side 2 feet are cut out from each corner. Then the sides are folded up to make an open box. Express the volume of the box as a function of the width (x).