Multiple Roots

A root $\alpha$ of $f(x)=0$ is said to have multiplicity $m$ if $(x-\alpha)^m$ divides $f(x)$ exactly but $(x-\alpha)^{m+1}$ does not — equivalently, $\alpha$ appears $m$ times among the $n$ roots counted in the Fundamental Theorem of Algebra. A root of multiplicity $1$ is called simple; one with multiplicity greater than $1$ is called multiple or repeated.

There is a clean test for multiple roots using calculus: if $\alpha$ is a root of $f(x)=0$ of multiplicity $m>1$, then $\alpha$ is also a root of the derivative equation $f'(x)=0$, with multiplicity exactly $m-1$; and if $\alpha$ is a simple root of $f$, then $f'(\alpha)\neq 0$. This makes sense geometrically — at a repeated root the graph of $f$ touches (rather than crosses) the $x$-axis, which is exactly the condition for a stationary point. Practically, this gives a two-line check: to see whether a specific candidate value is a repeated root of $f(x)=0$, verify $f(\alpha)=0$ and $f'(\alpha)=0$; to find an unknown repeated root, look among the common roots of $f(x)=0$ and $f'(x)=0$ (for instance, via their GCD).

Worked example. Show that $x=1$ and $x=2$ are both repeated roots of $f(x) = x^4 - 6x^3 + 13x^2 - 12x + 4 = 0$, and find their multiplicities. …