multiplicity of a root

How many times a factor (x - p) divides a polynomial -- a root of multiplicity 1 crosses the x-axis, a root of even multiplicity only touches it.

A root $p$ of a polynomial $f$ has multiplicity $m$ if $f(x)=(x-p)^m q(x)$ for some polynomial $q$ with $q(p)\ne0$ ; equivalently, $f(p)=f'(p)=\cdots=f^{(m-1)}(p)=0$ but $f^{(m)}(p)\ne0$ . A root of odd multiplicity changes sign there (the graph crosses); a root of even multiplicity does not (the graph only touches the axis) -- which is exactly why a bracketing method can silently miss a real root of even multiplicity, since no sign change ever appears around it.

Examples

$f(x)=(x-2)^2$ has a root of multiplicity 2 at $x=2$ : the graph touches the axis there but never crosses it, so no bracket $[a,b]$ around it can ever have $f(a)$ and $f(b)$ of opposite sign.