The graph of the rational function $\frac{q(x)}{x^6-4x^5+3x^3+x^2-1} $ has a horizontal asymptote. What is the largest possible degree of $q(x)$?
A rational function has a horizontal asymptote if the degree of the numerator and the degree of the denominator are equal or the degree of the numerator is less than the degree of the denominator. In this case, the degree of the denominator is 6, so the degree of the numerator must be less than or equal to 6. The largest possible degree of $q(x)$ is therefore 6. 