The graph of the function [math: f(x)] is shown here. The rule for [math: f(x)] is:

Question

BlackTom AI · Accepted Answer

To solve, we first identify the x-intercepts and their multiplicities from the graph, then match those with the polynomial factors provided.

Option A: f(x)=(x+2)(x−1)^2(3−x) expands to a product with a factor (3−x) which is the same as −(x−3). This flips the end behavior relative to a positive leading coefficient. The intercepts would be at x=−2, x=1 (double root), and x=3, but the overall sign pattern and end behavior implied by this form would differ from what the graph shows, so this option is unlikely despite having the correct roots and a double root at x=1.

Option B: f(x)=(x−2)(x+1)^2(x+3). This has zeros at x=−3 (simple), x=−1 (double), and x=2 (simple). The locations of the intercepts do not align with the graph, which shows intercepts near −2, 1, and 3, and the multiplicity at x=1 being even is essential for a bounce at the axis there. Therefore, B does not fit the graph.

Option C: f(x)=(x+2)(x−1)^2(x−3). This gives zeros at x=−2 (simple), x=1 (double), and x=3 (simple). The double root at x=1 explains a flattening or gentle bounce at the x-axis there, which matches the graph’s behavior around x=1. The simple roots at x=−2 and x=3 correspond to the graph crossing the axis at those points. Although the leading coefficient sign affects end behavior, the key features—two simple zeros at −2 and 3 with a double root at 1 and the corresponding crossing/bounce pattern—align with the graph’s zero structure.

Option D: f(x)=(x−2)(x−1)^2(x−3). This would place a simple zero at x=2 instead of at x=−2, shifting the left intercept, which does not agree with the graph’s intercept locations. The combination of roots does not match the depicted x-intercepts, so D is not correct.

In summary, the graph exhibits intercepts at x = −2 (simple), x = 1 (double, a bounce), and x = 3 (simple). Among the given forms, only option C has exactly this set of zeros with the correct multiplicities, matching the observed behavior at those x-values. The remaining options either place zeros at different x-values or assign an incorrect multiplicity, leading to mismatched crossing/bounce patterns on the graph.

The graph of the function [math: f(x)] is shown here. The rule for [math: f(x)] is:

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