Zero of a function

A zero of a function $f$ is an argument $a$ for which $f(a)=0$.

A root of a polynomial $f$ is a scalar $a$ satisfying $f(a)=0$.

For a polynomial $f$ over a field, $f(a)=0$ exactly when $X-a$ divides $f(X)$. Polynomial division gives $f(X)=(X-a)q(X)+f(a)$.

Multiplicity records how many times an object occurs when repetitions carry mathematical information.

The multiplicity of a root $a$ of a polynomial is the largest integer $m$ such that $(X-a{)}^m$ divides the polynomial.

A root is multiple when its multiplicity of a root is greater than one.

A common root of polynomials $f$ and $g$ is a scalar $a$ satisfying $f(a)=g(a)=0$.