1. **Problem Statement:** Given the polynomial function $$f(x) = (x - 7)^3 (x + 8)^2,$$ we need to find: (a) The real zeros and their multiplicities. (b) Whether the graph crosses or touches the x-axis at each zero. (c) The maximum number of turning points. (d) The end behavior of the graph. 2. **Finding Real Zeros and Multiplicities:** The zeros occur where each factor equals zero: - From $(x - 7)^3$, zero at $x = 7$ with multiplicity 3. - From $(x + 8)^2$, zero at $x = -8$ with multiplicity 2. 3. **Behavior at Each Zero:** - For multiplicity 3 (odd), the graph **crosses** the x-axis at $x=7$. - For multiplicity 2 (even), the graph **touches** but does not cross the x-axis at $x=-8$. 4. **Maximum Number of Turning Points:** The degree of $f(x)$ is $3 + 2 = 5$. The maximum number of turning points is one less than the degree: $$5 - 1 = 4.$$ 5. **End Behavior:** For large $|x|$, the function behaves like the leading term: $$f(x) \approx x^3 \cdot x^2 = x^5.$$ Since the leading coefficient is positive and degree is odd, as $x \to \infty$, $f(x) \to \infty$, and as $x \to -\infty$, $f(x) \to -\infty$. **Final answers:** (a) Real zeros: $7$ (multiplicity 3), $-8$ (multiplicity 2). (b) Crosses at $x=7$, touches at $x=-8$. (c) Maximum turning points: 4. (d) End behavior resembles $x^5$.