3.2: Polynomials--Classroom Activities

Pre-Calculus Readings and Assigments by Professor · LibreTexts

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    1. Let \(u(x)=-\frac{1}{5}(x-3)(x+1)(x+5)\) and \(v(x)=-\frac{1}{5}x^{2}(x-5)\).
      1. Find the zeros for \(u(x)\) and \(v(x)\).
      2. Graph \(u\) and \(v\) using a graphing calculator or other graphing tool. Did you identify the zeros correctly? How do you know?
      3. How are the graphs similar? How are they different?
      4. Using your graphing tool, zoom out to a much larger viewing window. What do you notice about the graphs now? How are they similar, and how are they different?
    2. Let \(y=1-2x^{4}+x^{3}\).
      1. How many terms does this polynomial have?
      2. What is the degree of this polynomial?
      3. What is the leading coefficient?
      4. What is the constant term?
      5. If \(x=100\), what is the value of each term? If \(x=100\), what is the value of the function?
      6. As \(x\) becomes really large, which term can be used as an approximation of the value of the function?
      7. Describe the end behavior of the function as \(x\rightarrow \infty\), and describe the end behavior of the function as \(x\rightarrow -\infty\).
    3. Let \(f(x)=-\frac{1}{5}(x-3)(x+1)(x+5)\).
      1. What is the degree of this polynomial?
      2. What is the leading coefficient?
      3. As \(x\) becomes really large, which term can be used as an approximation of the value of the function?
      4. Describe the end behavior of the function as \(x\rightarrow \infty\), and describe the end behavior of the function as \(x\rightarrow -\infty\).
    4. Let \(f(x)=\dfrac{1}{50,000}x^{3}+\dfrac{1}{2}x\).
      1. For small values of \(x\), which term of \(f\) is more important? Explain your answer.
      2. Graph \(y=f(x)\) using a regular viewing window. Does this graph appear linear? How does the appearance of this graph agree with your answer to part (a)?
      3. For large values of \(x\), which term of \(f\) is more important? Explain your answer.
      4. Using your graphing tool, zoom out to a much larger viewing window. What shape is the function now? Does the appearance of this graph agree with your answer to part (c)?
    5. Find:
      1. The end behavior of \(f(x)=3x^{2}-5x+7\)
      2. \(\lim\limits_{x\to \infty} (3x^{2}-5x+7)\)
      3. The end behavior of \(f(x)=7x^{2}-9x^{3}\)
      4. \(\lim\limits_{x\to -\infty} (7x^{2}-9x^{3})\)
    6. The town of Liddleville was founded in 1890. Its population \(y\) (in hundreds) was modeled, for \(t\) years since 1890, by \[y = 1 - 0.58t + 4.89t^2 - 1.872t^3 + 0.247t^4 - 0.0111t^5\]
      1. Graph the function using a graphing calculator or other graphing tool. Use an appropriate viewing window to make sure you understand what is happening with the function.
      2. What was the population of Liddleville when it was founded?
      3. When did the population of Liddleville reach zero? Give the year and the month.
      4. What was the largest population of Liddleville after 1890? When did Liddleville reach that population? Give the month and year.
      5. What population was predicted for 1898? Explain what this means.
    7. The volume, \(V\), in milliliters, of 1 kg of water as a function of temperature \(T\) is given, for \(0\leq T \leq 30\) degrees C, by \[V = 999.87 - 0.06426T + 0.0085143T^2 - 0.0000679T^3\]
      1. Graph \(V\).
      2. Describe the shape of your graph. Does \(V\) increase or decrease as \(T\) increases? Does the graph curve upward or downward? What does the graph tell us about how the volume varies with temperature?
      3. At what temperature does water have the maximum density? How does that appear on your graph?

    This page titled 3.2: Polynomials--Classroom Activities was last modified on Fri, 17 Jul 2026 16:24:17 GMT and is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Lynda Zenati.