Question

The function q is a polynomial of degree 3. If q(5) = 0, which of the following must be true? A) (x - 5) is a factor of q(x) B) q(0) = 5 C) q(x) = 5 D) q(3) = 0

Answer 1

The correct answer is that (x - 5) is a factor of q(x), as per the Factor Theorem that links zeros of a polynomial to its factors. None of the other options provided can be concluded from the information given that q(5) = 0.

Step-by-step explanation:

Given that the function q is a polynomial of degree 3 and q(5) = 0, it must be true that (x - 5) is a factor of q(x). This is due to the Factor Theorem, which states that if a polynomial f(x) has a zero at x = k, then (x - k) is a factor of f(x). Therefore, the correct answer is A) (x - 5) is a factor of q(x).

Option B) q(0) = 5 cannot be determined to be true just because q(5) = 0, as the value of the polynomial at x = 0 would depend on the specific constant term in the polynomial, which is not given. Option C) q(x) = 5 cannot be inferred, as q(x) could be any polynomial of degree 3 that has (x - 5) as a factor. Lastly, option D) q(3) = 0 is also not necessarily true, as the fact that q(5) = 0 does not provide any information regarding the value of q(x) at x = 3.

Answer 2

The option that is true for the function whose polynomial degree is 3 is (x - 5) is a factor of q(x).

Factorization of polynomial functions.

The factorization of polynomial functions is a way to simplify polynomial functions and get to know the roots of the polynomial and its factors.

In the given information;

The polynomial is said to be degree 3. If q(5) = 0. It means that x - 5 is a factor of q(x).

This is because for q(5) = 0, then x = 5

q(5) = 5 - 5

q(5) = 0

It cannot be q(0) = 5 because we are not given that information in the question. Also, the question talks about q(5) = 0, not q(x) = 5.