We are given the polynomial function:

$f(x) = (x - 3)^3(x + 1)^2$

Let's solve the problem step by step:

(a) List each real zero and its multiplicity

To find the real zeros of the function, we need to find the values of $x$ that make $f(x) = 0$.

1. $x - 3 = 0 \Rightarrow x = 3$ with multiplicity 3.
2. $x + 1 = 0 \Rightarrow x = -1$ with multiplicity 2.

Real zeros and their multiplicities:

• $x = 3$ with multiplicity 3.
• $x = -1$ with multiplicity 2.

(b) Determine whether the graph crosses or touches the​ x-axis at each​ x-intercept

The behavior of the graph at each x-intercept depends on the multiplicity of the zero:

• If the multiplicity is odd, the graph crosses the x-axis at that intercept.
• If the multiplicity is even, the graph touches the x-axis but does not cross it.

For $x = 3$ (multiplicity 3, odd): The graph crosses the x-axis.

For $x = -1$ (multiplicity 2, even): The graph touches the x-axis.

(c) Determine the maximum number of turning points on the graph

The maximum number of turning points of a polynomial function is given by $n - 1$, where $n$ is the degree of the polynomial.

The degree of the polynomial is the sum of the multiplicities of the factors:

$3 + 2 = 5$

So, the maximum number of turning points is:

$5 - 1 = 4$

(d) Determine the end behavior (power function)

To determine the end behavior, we need to consider the leading term of the polynomial for large values of $|x|$. We focus on the highest power of $x$ in each factor:

$(x - 3)^3 \text{ behaves like } x^3 \quad \text{and} \quad (x + 1)^2 \text{ behaves like } x^2$

Thus, the leading term is:

$x^3 \cdot x^2 = x^5$

The end behavior of the function resembles the power function:

$f(x) \sim x^5 \quad \text{for large values of} \ |x|.$

Final Answer for the Power Function:

$y = x^5$

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Do you want more details or have any questions?

Here are 5 related questions you could explore:

1. How does the degree of a polynomial affect the number of real zeros?
2. What happens to the graph at a zero with multiplicity higher than 3?
3. How can we find the turning points of a polynomial function graphically?
4. How do we distinguish between local and absolute extrema for polynomials?
5. How does the behavior of polynomials of odd degrees differ from those of even degrees?

Tip: To identify the number of turning points of a polynomial, focus on its degree and look for sign changes in the derivative.