Multiplicity of a root

For a nonzero polynomial over a field, the multiplicity of a root $a$ is the largest integer $m\ge 1$ such that $(X-a{)}^m$ divides it. Write $p(X)=(X-a{)}^{m}q(X)$ with $q(a)\ne 0$. For real polynomials, continuity keeps the sign of $q$ fixed sufficiently near $a$, so an odd integer $m$ makes the graph cross the axis and an even integer $m$ makes it touch without crossing.

The right-hand panel above replaces $(X-2{)}^2$ by $(X-2{)}^2-\epsilon$. For $\epsilon =0.04$, its two roots of a polynomial are $1.8$ and $2.2$; for $\epsilon =-0.04$, they are $2\pm 0.2i$. The unperturbed double root lies between these two cases. Small coefficient changes can therefore split a repeated root into distinct real roots or move it off the real axis, illustrating why repeated-root problems can be numerically sensitive.

A root of a polynomial is double when its multiplicity of a root is exactly two. Equivalently, $p(x)=(x-x_0{)}^{2}q(x)$ with $q(x_0)\ne 0$. A generic small perturbation splits it according to square-root splitting of a double polynomial root. Over a field of characteristic different from two, the derivative test is $p(a)=p^{\prime }(a)=0$ and $p^{\prime \prime }(a)\ne 0$. In characteristic two, use the Hasse derivative: a double root of $(X-a{)}^2$ has zero second ordinary formal derivative but nonzero second Hasse derivative.

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