Can the product of two polynomials result in a single term? [closed]

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.

Solution 1:

Yes. Take the polynomial $2X^2+X \in \mathbb{Z}_4[X]$. Then, $(2X^2+X)(2X^2+X)=X^2$.

If you are on an integral domain, this cannot happen. This is because the coefficients of the top degree and the lowest degree will be non-zero.

Therefore, if you are talking about polynomials with rational coefficients or integer coefficients, the answer is: No.

Bottom line: If you don't know what an integral domain is, you are probably talking about a polynomial with rational coefficients, to which the answer to your question is: No, that can't happen.

If you know what an integral domain is, then the answer to your question is: In general, yes, it can happen. But it can't happen in an integral domain.

Solution 2:

I assume that you speak about polynomials with real coefficients.

Suppose that the polynomials are $P(x)=ax^p+\cdots+bx^q$, $Q(x)=cx^r+\cdots+dx^s$, where

• $a,b,c,d\neq 0$
• $p>q\ge 0$ and $r>s\ge 0$
• Of course, the terms omitted between have intemediate degree.

Then, the product $P\cdot Q$ has at least two terms: $acx^{p+r}$ and $bdx^{q+s}$. Note that $ac$ and $bd$ are not zero.

Solution 3:

If you multiply two monomials you will get a monomial.

The only root of a monomial is x=0.
Polynomials (excluding monomials) have at least one non-zero root (possibly complex and possibly in addition to 0, but at least one non-zero root).
If we multiply one polynomial by another, the roots of the of the original polynomials are roots of the product.

If the original polynomial has a non-zero root, the product has non-zero roots, and is not a monomial.