Multiple Roots and the H.C.F. Method

A root $\alpha$ of $f(x)=0$ is called a multiple root of order $m$ (or a root of multiplicity $m$) if $f(x)=(x-\alpha)^mg(x)$ with $g(\alpha)\neq0$. Differentiating this factorisation shows $f'(x)=(x-\alpha)^{m-1}\big[mg(x)+(x-\alpha)g'(x)\big]$, and since the bracket does not vanish at $x=\alpha$ (it equals $mg(\alpha)\neq0$), the root $\alpha$ survives in $f'(x)=0$ with multiplicity exactly $m-1$ -- one less than it had in $f$.

This single fact turns the problem of finding all the multiple roots of $f(x)=0$ into a single computation: every multiple root of $f$ is a root of $\mathrm{HCF}\big(f(x),f'(x)\big)$, and conversely every root of this H.C.F. is a multiple root of $f$, now with multiplicity exactly one less. So computing $d(x)=\mathrm{HCF}(f,f')$ by the ordinary Euclidean algorithm for polynomials (repeated division with remainder, dropping any convenient constant factor along the way, since a constant multiple never changes an H.C.F.) identifies every re …