Does multiplying polynomials ever decrease the number of terms? Let $p$ and $q$ be polynomials (maybe in several variables, over a field), and suppose they have $m$ and $n$ non-zero terms respectively. We can assume $m\leq n$. Can it ever happen that the product $p\cdot q$ has fewer than $m$ non-zero terms?
I ask this because I vaguely recall seeing a positive answer in book somewhere (probably about computation or algorithms since the polynomials were unwieldy). If anyone knows what book this is from it would be much appreciated.

Here's an elementary example. Start with the well-known identity $x^n - 1 = (x-1) (x^{n-1} + x^{n-2} + \ldots + x + 1)$. If $n$ is odd, we can factor $x^n+1$ in a similar way by flipping the signs: $x^n + 1 = (x+1) (x^{n-1} - x^{n-2} + \ldots - x + 1)$. Now mix and match the two:
$$\begin{align*}
  x^{2n} - 1 &= (x^n - 1) (x^n + 1) \\
  &= (x-1) (x^{n-1} + x^{n-2} + \ldots + x + 1) (x+1) (x^{n-1} - x^{n-2} + \ldots - x + 1) \\
  &= (x+1) (x^{n-1} + x^{n-2} + \ldots + x + 1) (x-1) (x^{n-1} - x^{n-2} + \ldots - x + 1) \\
  &= (x^n + 2x^{n-1} + 2x^{n-2} + \ldots + 2x + 1) (x^n - 2x^{n-1} + 2x^{n-2} - \ldots + 2x - 1)
\end{align*}$$
I don't see an obvious generalization to even values of $n$.
