Final answer:
The graph displays roots at x = 5, x = 2, and x = 0, matching the factors (x - 5), (x - 2), and (x¹). So, the correct is option 3) f(x) = (x - 5)(x - 2)(x¹).
Explanation:
The correct is option 3) f(x) = (x - 5)(x - 2)(x¹). The given graph exhibits three distinct roots at x = 5, x = 2, and x = 0. To decipher the correct representation, we examine the factors in each option. Option 3, f(x) = (x - 5)(x - 2)(x¹), accurately aligns with the observed roots. The factor (x - 5) corresponds to the root at x = 5, (x - 2) to the root at x = 2, and (x¹) to the root at x = 0. The absence of additional factors ensures that these are the only roots, consistent with the graph.
In contrast, other options either include unnecessary factors (such as x¹⁰ in option 1 and x - 10 in option 4) or lack necessary factors corresponding to observed roots. Option 3, with its three linear factors, is the simplest polynomial that captures the essential characteristics of the graph.
Understanding polynomial graphs involves analyzing roots and their corresponding factors. In this case, the roots at x = 5, x = 2, and x = 0 guide us to option 3 as the most accurate representation of the given graph.