Minimize the number of roots

Algebra Level 5

Suppose $p(x)$ and $q(x)$ are polynomials of degree $100$ with complex coefficients, having no common zeroes. Find the smallest possible total number of complex zeroes of the polynomials $p,$ $q,$ and $p-q,$ counted without multiplicity.

Details and assumptions

When the zeroes are counted without multiplicity, the polynomial $x^2(x-1)^3$ has two zeroes: $x=0$ and $x=1.$