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Section 2.2 – Polynomial Functions of Higher Degree

Section Details.
  • Definition of a polynomial function and its properties such as the degree, leading coefficient, and constant term
  • Graphing a polynomial function
  • Finding the end behavior of a polynomial function
  • Finding the zeros (\(x\)-intercepts) of a polynomial function
  • Using the multiplicity of a zero to determine whether the graph crosses or just touches the \(x\)-axis at that \(x\)-value


Practice Problems


Directions. The following are review problems for the section. We recommend you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it. 
 
  1. For the given polynomial functions, determine the end behavior of the graph.
    1. \(f(x)=-4x^8+7x^5-1\)

      As \(x\to -\infty\), \(f(x)\to -\infty\) and as \(x\to \infty\), \(f(x)\to -\infty\).


      To see the full video page and find related videos, click the following link.
      WIR3 20B M150 V4


    2. \(g(x)=11x^7+6x-10\)

      As \(x\to -\infty\), \(g(x)\to -\infty\) and as \(x\to \infty\), \(g(x)\to \infty\).


      To see the full video page and find related videos, click the following link.
      WIR3 20B M150 V5


    3. \(h(x)=14x^2-9x^3+6\)

      As \(x\to -\infty\), \(h(x)\to \infty\) and as \(x\to \infty\), \(h(x)\to -\infty\).


      To see the full video page and find related videos, click the following link.
      WIR3 20B M150 V6


  2. Find the zeros and their multiplicities for the following functions, and then determine the end behavior and maximum number of turning points.
    1. \(k(x)=2x^3+5x^2+9x\)

      \(x=0\) odd multiplicity
      As \(x\to -\infty\), \(k(x)\to -\infty\) and as \(x\to \infty\), \(k(x)\to \infty\)
      Maximum number of turning points is 2.


      To see the full video page and find related videos, click the following link.
      WIR3 20B M150 V7

    2. \(g(x)=-x^3+8x^2-16x\)

      \(x=0\) odd multiplicity,
      \(x=4\) even multiplicity
      As \(x\to -\infty\), \(g(x)\to \infty\) and as \(x\to \infty\), \(g(x)\to -\infty\)
      Maximum number of turning points is 3.


      To see the full video page and find related videos, click the following link.
      WIR3 20B M150 V8

    3. \(h(x)=4x^4+12x^3+9x^2\)

      \(x=0\) even multiplicity,
      \(x=-\dfrac{3}{2}\) even multiplicity
      As \(x\to -\infty\), \(h(x)\to \infty\) and as \(x\to \infty\), \(h(x)\to \infty\)
      Maximum number of turning points is 3.


      To see the full video page and find related videos, click the following link.
      WIR3 20B M150 V9

  3. Match the function \(g(x)=-x^3+8x^2-16x\) to its graph.
    1. The correct graph is d)


      To see the full video page and find related videos, click the following link.
      WIR3 20B M150 V10