1999 AHSME Problem 12

What is the maximum number of points of intersection of the graphs of two different fourth degree polynomial functions y = p(x) and y = q(x), each with leading coefficient 1?

\textrm{(A)} \ 1 \qquad \textrm{(B)} \ 2 \qquad \textrm{(C)} \ 3 \qquad \textrm{(D)} \ 4 \qquad \textrm{(E)} \ 8

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.