smc

The intersections of the two polynomials, $p(x)$ and $q(x)$, are precisely the roots of the equation $p(x)=q(x)\to p(x)-q(x)=0$. Since the leading coefficients of both polynomials are $1$, the degree of $p(x)-q(x)=0$ is at most three, and the maximum point of intersection is three, because a third degree polynomial can have at most three roots. Thus, the answer is $\boxed{3}$.