1. **Problem Statement:** Identify the polynomial equation from the given options based on the graph's behavior at its roots. 2. **Given Information:** The graph has roots at $x=1$, $x=2$, and $x=4$. - At $x=1$, the curve crosses the x-axis with a steep slope, indicating a root of multiplicity 1. - At $x=2$, the curve touches the x-axis and turns around, indicating a root of multiplicity 2. - At $x=4$, the curve crosses the x-axis again with a steep slope, indicating a root of multiplicity 1. 3. **Key Concept:** The multiplicity of a root affects the graph's behavior at that root: - Odd multiplicity roots cross the x-axis. - Even multiplicity roots touch the x-axis and turn around. 4. **Analyze each option:** - $y = (x - 1)(x - 2)^2(x - 4)$: multiplicities 1, 2, 1 respectively. - $y = (x - 1)^3(x - 2)^2(x - 4)$: multiplicities 3, 2, 1. - $y = (x - 1)^3(x - 2)^2(x - 4)^3$: multiplicities 3, 2, 3. - $y = (x - 1)(x - 2)^2(x - 4)^3$: multiplicities 1, 2, 3. 5. **Match with graph behavior:** - At $x=1$, the root is simple (multiplicity 1), so options with $(x-1)^3$ are incorrect. - At $x=4$, the root is simple (multiplicity 1), so options with $(x-4)^3$ are incorrect. 6. **Conclusion:** The only option matching the graph's root behavior is: $$y = (x - 1)(x - 2)^2(x - 4)$$ This polynomial has roots at $x=1$ and $x=4$ with multiplicity 1 (crossing the x-axis) and root at $x=2$ with multiplicity 2 (touching and turning). **Final answer:** $$y = (x - 1)(x - 2)^2(x - 4)$$