Question

What is the polynomial function that satisfies the given properties: Roots at 0, 4, and -4 with multiplicities 2, 1, and 1, respectively?

A) f(x) = x^2(x - 4)(x + 4)
B) f(x) = x^2(x - 4)^2(x + 4)
C) f(x) = x^2(x - 4)(x + 4)^2
D) f(x) = x^2(x - 4)(x + 4)

Answer 1

The correct polynomial function is f(x) = x^2(x - 4)(x + 4), reflecting the specified roots and their multiplicities.

Step-by-step explanation:

The polynomial function that satisfies the given properties with roots at 0 with multiplicity 2, and roots at 4 and -4 with multiplicities 1, respectively, can be constructed by multiplying the binomials corresponding to each root raised to the power of their multiplicity. Since the root at 0 has a multiplicity of 2, we include a factor of x^2. The roots at 4 and -4 each have a multiplicity of 1, so we include the factors (x - 4) and (x + 4) without additional powers. Therefore, the correct polynomial function is f(x) = x^2(x - 4)(x + 4).