For the polynomial function $f (x) = 7 (x - 9) (x + 2)^{2}$ answer the following questions.
(a) List each real zero and its multiplicity.
(b) Determine whether the graph crosses or touches the $x$ -axis at each $x$ -intercept.
(c) Determine the maximum number of turning points on the graph.
(d) Determine the end behavior; that is, find the power function that the graph of $f$ resembles for large values of $| x |$ .
(a) Find any real zeros of $f$ . Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
A. The real zero of $f$ is $◻$ with multiplicity
(Type an exact answer, using radicals as needed. Type integers or fractions.)
B. The smallest zero of $f$ is with multiplicity The largest zero of $f$ is with multiplicity
(Type an exact answer, using radicals as needed. Type integers or fractions.)
C. The smallest zero of $f$ is $◻$ with multiplicity The middle zero of $f$ is with multiplicity The largest zero of $f$ is with multiplicity (Type an exact answer, using radicals as needed. Type integers or fractions.)
D. There are no real zeros.

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Final Answer: The real zeros of the function $f (x) = 7 (x - 9) (x + 2)^{2}$ are $x = - 2$ with multiplicity 2 and $x = 9$ with multiplicity 1. In Latex format, the answer is $x = - 2 with multiplicity 2 and x = 9 with multiplicity 1$ .

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Step 1 ：The real zeros of a polynomial function are the x-values for which the function equals zero. In this case, the function will be zero when either $(x - 9)$ or $(x + 2)^{2}$ equals zero. So, we need to solve these two equations to find the real zeros.

Step 2 ：The solutions to the equations are x = 9 and x = -2. However, since $(x + 2)^{2}$ is squared, the zero x = -2 has a multiplicity of 2, meaning it is a root of the polynomial twice. The zero x = 9 has a multiplicity of 1, as it is not raised to any power.

Step 3 ：Final Answer: The real zeros of the function $f (x) = 7 (x - 9) (x + 2)^{2}$ are $x = - 2$ with multiplicity 2 and $x = 9$ with multiplicity 1. In Latex format, the answer is $x = - 2 with multiplicity 2 and x = 9 with multiplicity 1$ .

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