Final answer:
Statement D is true because (x² - 4) is a factor of f(x). We cannot confirm statements A and B without additional information, and statement E is false since the sign must change at least three times. Statement C is incorrect because (x - 5) is not a factor.
Step-by-step explanation:
If the function f(x) has a leading term of 3x⁴ and zeros at x=2, x=-2, and x= -5, we can determine some possible characteristics of f(x).
First, since x=2 and x=-2 are zeros, we know that (x - 2) and (x + 2) are factors, which means (x² - 4) is a factor of f(x) due to it being the product of those factors. Thus, statement D is true.
Given that x= -5 is a zero, (x + 5) is a factor. After adjusting for the sign, we see that (x - 5) is not a factor, making statement C incorrect.
Because f(x) is a quartic polynomial and we already have three real zeros (including a pair of opposite sign that creates a quadratic factor), we may have one additional real zero or two complex conjugate zeros. However, without further information, we cannot confirm if 3 - 2√10 is a zero, so statement A cannot be confirmed solely on the data provided.
Regarding the constant term of f(x), without the explicit polynomial or the product of all zeros (including multiplicities), we cannot determine if the constant term is 4. Hence, we cannot confirm statement B.
Considering the given zeros, the function f(x) will change signs each time it passes through a zero, therefore, changes its sign at least three times. This invalidates statement E which says the sign of f(x) changes twice.