Final answer:
When a polynomial function of degree 3 has a root (x-value) of 5 (q(5) = 0), it means that (x - 5) is a factor of the polynomial. By substituting values into the polynomial equation, we can determine which statement is true. In this case, q(7) must be equal to 0. Option D is correct.
Step-by-step explanation:
When a polynomial function of degree 3 has a root (x-value) of 5 (q(5) = 0), it means that (x - 5) is a factor of the polynomial. So, q(x) can be written as q(x) = (x - 5)(ax^2 + bx + c), where a, b, and c are coefficients.
To determine which statement must be true, we can substitute the given values into the polynomial equation:
a) q(0) = 0: Substituting x = 0, q(0) = (0 - 5)(a(0)^2 + b(0) + c) = -5c. This statement may or may not be true, as it depends on the value of c. It is not necessarily true because c could be any number.
b) q(3) = 0: Substituting x = 3, q(3) = (3 - 5)(a(3)^2 + b(3) + c) = (3 - 5)(9a + 3b + c) = -2(9a + 3b + c) = -18a - 6b - 2c. This statement may or may not be true, as it depends on the values of a, b, and c. However, it is possible for q(3) to be equal to 0.
c) q(5) = 1: Substituting x = 5, q(5) = (5 - 5)(a(5)^2 + b(5) + c) = 0. This statement is not true, as q(5) cannot be 1.
d) q(7) = 0: Substituting x = 7, q(7) = (7 - 5)(a(7)^2 + b(7) + c) = 0. This statement is true, as q(7) can be equal to 0.
Therefore, the correct answer is d) q(7) = 0.