1. **Problem Statement:** Given the polynomial function $$f(x) = 6 (x^2 + 4)^2 (x - 3)^3,$$ answer the following: (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 of the graph. 2. **Step (a): Find real zeros and multiplicities** - The zeros come from factors set to zero. - For $$x^2 + 4 = 0,$$ solve $$x^2 = -4,$$ which has no real solutions. - For $$x - 3 = 0,$$ solve $$x = 3,$$ which is a real zero. - The multiplicity of zero $$x=3$$ is 3 (from the exponent on $$(x-3)^3$$). 3. **Step (b): Crossing or touching x-axis** - If the multiplicity is odd, the graph crosses the x-axis at that zero. - If the multiplicity is even, the graph touches the x-axis and turns around. - Here, multiplicity 3 is odd, so the graph crosses the x-axis at $$x=3$$. 4. **Step (c): Maximum number of turning points** - The degree of $$f(x)$$ is $$2 \times 2 + 3 = 4 + 3 = 7$$. - Maximum turning points for a polynomial of degree $$n$$ is $$n-1$$. - So, maximum turning points = $$7 - 1 = 6$$. 5. **Step (d): End behavior** - For large $$|x|,$$ the dominant term is the highest degree term. - Expand the leading terms: $$(x^2)^2 = x^4$$ and $$(x-3)^3 \approx x^3$$ for large $$x$$. - So, $$f(x) \approx 6 x^4 x^3 = 6 x^7$$. - 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 zero: $$3$$ with multiplicity $$3$$. (b) Graph crosses the x-axis at $$x=3$$. (c) Maximum turning points: $$6$$. (d) End behavior resembles $$6 x^7$$.