# Proof for an ACT math question?

A polynomial in x has m nonzero terms. Another polynomial in x has n nonzero terms, where m is less than n. These polynomials are multiplied and all like terms are combined. The resulting polynomial has a maximum of how many nonzero terms? How would you prove that the answer is mn?

Each non-zero coefficient in the product must be the result of at least one pair from the original polynomials so the number cannot exceed $mn$.
$mn$ is a possible result, for example from $(x^{(m-1)n}+x^{(m-2)n}+\cdots+x^{2n}+x^{n}+1)(x^{n-1}+x^{n-2}+\cdots+x^2+x^1+1)$