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 $\mathrm{f}$. 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 $\square$ 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 $\mathrm{f}$ is with multiplicity (Type an exact answer, using radicals as needed. Type integers or fractions.) C. The smallest zero of $f$ is $\square$ 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.
Solution
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 \(\boxed{x = -2 \text{ with multiplicity 2 and } x = 9 \text{ with multiplicity 1}}\).
