Root of a polynomial

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

A positive root of a real polynomial $p$ is a real number $r>0$ for which $p(r)=0$. Zero is excluded. A count of distinct positive roots counts each such number once; a count with multiplicity of a root counts a root of order $m$ exactly $m$ times. Multiplication by a power of $x$ does not change either positive-root count.

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 $f(a)=g(a)=0$.