Final Answer:
The polynomial that meets the given conditions is f(x) = 3x⁵ - 9x³.
Step-by-step explanation:
In order to construct a polynomial that satisfies the given conditions, we need to analyze the provided options. The original polynomial is given as f(x) = 3x⁵. By examining the options, we observe a common factor of 3x⁵ in all the given polynomials. This common factor indicates that the leading term of our polynomial should be 3x⁵.
Moving on to the options, we see that the second term in each polynomial involves powers of x³. The additional terms in the polynomial should contribute to the desired conditions. Among the given choices, option C, f(x) = 3x⁵ - 9x³, stands out as the correct answer. This choice not only includes the required leading term but also subtracts a term with a coefficient of 9x³, meeting the given conditions specified in options B, C, and D.
To validate this choice, let's perform a quick check. If we simplify f(x) = 3x⁵ - 9x³, we get the required leading term of 3x⁵, and the subtraction of 9x³ satisfies the conditions given in options B, C, and D. Therefore, option C, f(x) = 3x⁵ - 9x³, is the appropriate polynomial that meets the specified criteria.