Question: Which of the following polynomials matches the graph? Graph: A blue cubic-like curve with left end down and right end up, touching the x-axis at x = -4 (double root) and crossing the x-axis at x = 1 (single root). The graph is in the top-left. A f(x) = (x - 4)(x + 1)^2 B f(x) = - (x - 1)(x + 4)^2 C f(x) = (x - 1)(x + 4)^2 D f(x) = (x - 4)^2(x + 1)

1. **State the problem:** We need to identify which polynomial matches the given graph. 2. **Analyze the graph:** - The graph touches the x-axis at $x = -4$ with a double root (meaning the graph just touches and turns around). - The graph crosses the x-axis at $x = 1$ with a single root (meaning it passes through the axis). - The left end of the graph goes down and the right end goes up, indicating the leading coefficient is positive. 3. **Recall root behavior:** - A double root at $x = a$ means the factor $(x - a)^2$. - A single root at $x = b$ means the factor $(x - b)$. 4. **Check each option:** - Option A: $f(x) = (x - 4)(x + 1)^2$ - Roots at $x=4$ (single) and $x=-1$ (double), which does not match the graph. - Option B: $f(x) = - (x - 1)(x + 4)^2$ - Roots at $x=1$ (single) and $x=-4$ (double), correct roots. - Negative leading coefficient (due to $-$), so ends go up-down, which contradicts the graph. - Option C: $f(x) = (x - 1)(x + 4)^2$ - Roots at $x=1$ (single) and $x=-4$ (double), correct roots. - Positive leading coefficient, so ends go down-up, matching the graph. - Option D: $f(x) = (x - 4)^2(x + 1)$ - Roots at $x=4$ (double) and $x=-1$ (single), does not match the graph. 5. **Conclusion:** Option C matches the graph. **Final answer:** $$f(x) = (x - 1)(x + 4)^2$$