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Reed College Department of Mathematics & Statistics

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Worksheet Polynomial Functions

6.

Determine the end behavior of the following polynomial function:
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\begin{equation*} f(x) = -4 x^{8} + 5 x^{3} - 12 x^{12} - 60 x^{10} - 35 \end{equation*}
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The leading term of the polynomial is
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The degree of \(f(x)\) is
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The leading coefficient is
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The end behavior of the polynomial \(f(x)\) is of the form:
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where:
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Answer 1.
\(-12x^{12}\)
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Answer 2.
\(12\)
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Answer 3.
\(-12\)
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Answer 4.
\(\text{D}\)
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7.

Determine the end behavior of the following polynomial function:
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\begin{equation*} f(x) = 2 + 20 x^{4} + 17 x^{9} - 6 x^{14} \end{equation*}
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The leading term of the polynomial is
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The degree of \(f(x)\) is
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The leading coefficient is
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The end behavior of the polynomial \(f(x)\) is of the form:
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where:
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Answer 1.
\(-6x^{14}\)
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Answer 2.
\(14\)
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Answer 3.
\(-6\)
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Answer 4.
\(\text{D}\)
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8.

Given \(f(x) = (x-1)(x-8)(x + 3)\text{,}\) find the roots in increasing order.
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The roots are , , and .
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To the left of the first root, is the graph of \(f(x)\) above or below the x-axis? Answer above or below: .
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Between the first two roots, is the graph of \(f(x)\) above or below the x-axis? Answer above or below: .
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Between the last two roots, is the graph of \(f(x)\) above or below the x-axis? Answer above or below: .
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After the last root, is the graph of \(f(x)\) above or below the x-axis? Answer above or below:
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Answer 1.
\(-3\)
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Answer 2.
\(1\)
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Answer 3.
\(8\)
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Answer 4.
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Answer 5.
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Answer 6.
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Answer 7.
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15.

For \(f(x) = {2x^{3}\mathopen{}\left(2x-4\right)}\text{,}\) determine the `x`-intercepts and the end behavior. Enter intercepts as a comma separated list of points.
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Intercepts :
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Inputting the function into the Desmos graphing tool below may help investigate the function.
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Answer 1.
\(\left(0,0\right), \left(2,0\right)\)
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Answer 2.
\(\infty \)
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Answer 3.
\(\infty \)
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16.

For \(f(x) = {-3x^{3}\mathopen{}\left(-2x-6\right)}\text{,}\) determine the `x`-intercepts and the end behavior. Enter intercepts as a comma separated list of points.
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Intercepts :
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Inputting the function into the Desmos tool below may help investigate the function.
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Answer 1.
\(\left(0,0\right), \left(-3,0\right)\)
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Answer 2.
\(\infty \)
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Answer 3.
\(\infty \)
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17.

For \(f(x) = {\left(3-x\right)\mathopen{}\left(4x-1\right)\mathopen{}\left(x^{2}-4\right)}\text{,}\) determine the `x`-intercepts and the end behavior. Enter intercepts as a comma separated list of points.
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Intercepts :
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Inputting the function into the Desmos tool below may help investigate the function.
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Answer 1.
\(\left(3,0\right), \left(0.25,0\right), \left(2,0\right), \left(-2,0\right)\)
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Answer 2.
\(-\infty \)
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Answer 3.
\(-\infty \)
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18.

For \(f(x) = {3x^{3}\mathopen{}\left(2x-6\right)}\text{,}\) determine the `x`-intercepts and the end behavior. Enter intercepts as a comma separated list of points.
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Intercepts :
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Inputting the function into the Desmos tool below may help investigate the function.
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Answer 1.
\(\left(0,0\right), \left(3,0\right)\)
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Answer 2.
\(\infty \)
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Answer 3.
\(\infty \)
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