Polynomial Graph 30Aa9B

1. **State the problem:** We need to determine which graph correctly represents the polynomial $$f(x) = 2x^3(x - 2)^2$$. 2. **Analyze the polynomial:** - The polynomial is $$f(x) = 2x^3(x - 2)^2$$. - It is a product of a cubic term $$2x^3$$ and a squared term $$(x - 2)^2$$. 3. **Find the roots and their multiplicities:** - Root at $$x=0$$ with multiplicity 3 (odd multiplicity). - Root at $$x=2$$ with multiplicity 2 (even multiplicity). 4. **Behavior at roots:** - At $$x=0$$ (odd multiplicity 3), the graph crosses the x-axis and flattens (because multiplicity > 1). - At $$x=2$$ (even multiplicity 2), the graph touches the x-axis and bounces off (does not cross). 5. **End behavior:** - Leading term is $$2x^3 imes x^2 = 2x^5$$ (degree 5, positive leading coefficient). - For large $$x o +$$, $$f(x) o +$$ (right tail up). - For large $$x o -$$, $$f(x) o -$$ (left tail down). 6. **Local extrema:** - The graph should have a local maximum near $$x=1.3$$ and a local minimum near $$x=2$$ (touching the axis). 7. **Match with given graph descriptions:** - The first graph matches all these features: left tail down, right tail up, crosses x-axis at 0 with flattening, touches and bounces at 2, local max near 1.3. - The other graphs do not match these root behaviors or end behaviors. **Final answer:** The first graph is the correct sketch of $$f(x) = 2x^3(x - 2)^2$$.