# 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 $$x\to \infty$$, $$f(x)\to -\infty$$.

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

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

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

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

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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.

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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.

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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.

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

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