The function q(x) is a polynomial. If q(–12)=–3 and q(–8)=3, can you conclude that q(x) is ever equal to 0?

Yes, we can conclude that q(x) is ever equal to 0. This is because of the Intermediate Value Theorem, which states that if a function is continuous on a closed interval [a, b] and k is any number between f(a) and f(b), then there is at least one number c in the interval (a, b) such that f(c) = k.

In this case, the function q(x) is a polynomial, and polynomials are continuous everywhere. The values given are q(–12)=–3 and q(–8)=3. So, we have a closed interval [-12, -8] and 0 is a number between -3 and 3. Therefore, there must be some number c in the interval (-12, -8) such that q(c) = 0.