Is it possible when multiplying two polynomials that, after collecting similar terms, all terms vanish? Algebra by Gelfand poses this question with the remarkably unhelpful answer of:

No. Probably this problem seems silly; it is clear that it cannot happen. If you think so, please reconsider the problem several years from now.

I'm sure the mathematical wit is just lost on me, but since you can lose some terms from multiplying, it doesn't seem too far fetched through some mathematical wizardry that there are cases where they all vanish. So why is this?
If you are working with polynomials over a ring with zero divisors, such as $\mathbb{Z}/4\mathbb{Z}$, then it is possible for the product of two polynomials to vanish.  This may be what Gelfand is coyly alluding to.  But in the ordinary sense of polynomials with rational, real, or complex coefficients, the degree of the product is the sum of the degrees of the polynomials being multiplied together.
Sure, if the polynomials are piecewise defined.  Let p(x)={x^2, x>0; 0,x&lt;0}, q(x)={x^3,x&lt;=0;0,x>0}.  But these wouldn't be polynomials in the strict definition.