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.

Just look at the highest-order coefficient $-$ it can't be zero, because it is the product of the highest-order coefficients of $p$ and $q$.

Consider the zero polynomial $p(x)=0$, then certainly multiplying any real polynomial by $p(x)$ yields $0$.

For a polynomial with $\deg\geq0$, over some field $\mathbb{F}$, this is not possible.

Consider the general polynomial of the form:

$$p(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_2x^2+a_1x+a_o$$

For $a_n$ having real or complex coefficients.

Clearly, as you multiply two polynomials, the $\deg$ of each term increases, except for the constant case, which yields a proportional polynomial.

So we can simply say, for all $x\in \mathbb{R}$, such that two real polynomials $p(x)g(x)=0$ implies $p(x) \ or \ g(x)=0$.

Hint: Try to consider the product of two linear terms: $(ax+b)(cx+d)$ where $a,b,c,d$ are real numbers with $a,c$ being nonzero. When expanding and simplifying (collecting like terms), for which terms is it possible for those terms to have coefficients of zero?

After answering this, can you generalize this to the product of any two non-zero polynomials of any degree?

EDIT: There is the slight issue of distinguishing polynomials and polynomial functions. One may consider two polynomials as equal if and only if their corresponding coefficients are equal. On the other hand, one may consider two polynomials as equal if and only if they are equal as functions (i.e. polynomials $p,q$ are equal if and only if $p(x) = q(x)$ are equal for all $x$).

It turns out that for polynomials in $\mathbb{R}$, we don't need to worry about such a distinction (although you would need to if you were dealing with polynomials over a ring as Barry Cipra mentioned in his answer). It may be a good exercise to show that two polynomials are equal over $\mathbb{R}$ if and only if they are equal as polynomial functions.

If our two polynomials are $P(x_1,x_2,\cdots, x_n)$ and $Q(x_1,x_2,\cdots, x_n)$, this means $P(x_1,x_2,\cdots, x_n)\cdot Q(x_1,x_2,\cdots, x_n) = 0.$ If neither of these polynomials are the zero polynomial, there must exist some set $(x_1,x_2,\cdots, x_n)$ such that neither polynomial evaluates to $0$ there. Then $P\cdot Q \neq 0$.