= Multiplicity of a root
For a nonzero <polynomial> over a <field>, the multiplicity of a root $a$ is the largest <integer> $m\geq1$ such that $(X-a)^m$ divides it. Write $p(X)=(X-a)^m q(X)$ with $q(a)\ne0$. 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 $(X-2)^2-\varepsilon$. For $\varepsilon=0.04$, its two <roots of a polynomial> are $1.8$ and $2.2$; for $\varepsilon=-0.04$, they are $2\pm0.2i$. 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.
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