TITLE: Can the product of two polynomials result in a single term?
QUESTION [11 upvotes]: Assume that the polynomials that we multiply consist of more than one term.
I don't think we can get a result containing only a single term, but I don't know how to prove it.

REPLY [2 votes]: It fails iff the coefficient ring has zero-divisors, i.e. $\,ab=0\,$ for $\,a,b\neq 0.\,$ Indeed if so then
$$ (ax+b)\,(bx+a)\, =\, (a^2\!+b^2)\, x$$
Conversely, if there are no zero-divisors, then the coefficients of the least and greatest degree terms of the product do not vanish (as above), yielding at least two terms in the product.