By assuming that we are talking about polynomials with real coefficients. Suppose that the polynomials are,
$\mathrm{P(x)\; =\; axp+\; ...\; +\; bxq}$
$\mathrm{Q(x)\; =\; cxr+\; ...\; +\; dxs}$
Where . . . $\mathrm{Q(x)\; =\; cxr+\; ...\; +\; dxs}$

$$
a , b , c , d ≠ 0 
$p>q\ge 0\mathrm{}$
and
$$
r > s ≥ 0 
$$
p + r > q + s  Of course, the terms omitted between have intermediate degree.
Then, the product
$\mathrm{P.Q}$
has at least two terms
acx^{p+r}
and
bdx^{q+s}
where ac and bd are not zero.
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