Can we determine the divisibility of a polynomial by another polynomial from their roots?

Question

I had this small doubt. Is a polynomial divided by another polynomial if the set of roots of one polynomial is the subset of the set of roots of the other polynomial?

Answer 1

There are two main points you forget to refer to -

• multiplicity of a root - For example, $p(x)=x-1$ has a root $1$ with multiplicity $1$, but $p(x)=(x-1)^2$ has multiplicity $2$.

• reducibility of the polynomial - For example, the polynomial $x^2+1$ is irreducible (and has no roots) over the real numbers.

You can fix your claim in the following manner -

If $p(x)$ and $q(x)$ are two polynomials such that:

• both $p(x)$ and $q(x)$ are completely reducible to linear factors

• every root of $p(x)$ is a root of $q(x)$, where the multiplicity in $q$ is higher

Then $p(x)$ is dividing $q(x)$.