1. **State the problem:** We have the polynomial function $$f(x) = \frac{1}{3} (3x^2 + 8)^2 (x^2 + 9)$$ and need to find its real zeros and their multiplicities, determine if the graph crosses or touches the x-axis at each zero, find the maximum number of turning points, and describe the end behavior. 2. **Find real zeros:** Real zeros occur where $$f(x) = 0$$. Since $$f(x)$$ is a product, zeros come from factors equal to zero. 3. **Analyze each factor:** - $$3x^2 + 8 = 0 \implies x^2 = -\frac{8}{3}$$ which has no real solutions because $$x^2$$ cannot be negative. - $$x^2 + 9 = 0 \implies x^2 = -9$$ also no real solutions. 4. **Conclusion on zeros:** Neither factor has real roots, so $$f(x)$$ has no real zeros. 5. **Multiplicity:** Since there are no real zeros, multiplicities do not apply. 6. **Graph behavior at zeros:** No real zeros means no x-intercepts, so the graph neither crosses nor touches the x-axis. 7. **Maximum number of turning points:** The degree of $$f(x)$$ is the sum of degrees of factors: - $$3x^2 + 8$$ is degree 2, squared gives degree 4. - $$x^2 + 9$$ is degree 2. - Total degree $$= 4 + 2 = 6$$. Maximum turning points for a polynomial of degree $$n$$ is $$n - 1$$, so here maximum turning points $$= 6 - 1 = 5$$. 8. **End behavior:** For large $$|x|$$, the dominant term is the highest degree term. Expand leading terms: - $$3x^2 + 8 \approx 3x^2$$ so $$ (3x^2)^2 = 9x^4$$ - $$x^2 + 9 \approx x^2$$ So $$f(x) \approx \frac{1}{3} \times 9x^4 \times x^2 = 3x^6$$. Thus, for large $$|x|$$, $$f(x)$$ behaves like $$3x^6$$. **Final answers:** (a) There are no real zeros. (b) The graph does not cross or touch the x-axis. (c) Maximum number of turning points is 5. (d) End behavior resembles $$3x^6$$.