What is Derivative, multiplicity?

Example 8.12 (Computing a multiplicity)

What is the multiplicity of the root $2$ in $P = X^4 - 5X^3 + 6X^2 + 4X - 8$? Evaluate the successive derivatives at $2$:

(with $P' = 4X^3 - 15X^2 + 12X + 4$, $P'' = 12X^2 - 30X + 12$, $P''' = 24X - 30$). Three vanishing values then a nonzero one: multiplicity exactly $3$. Dividing, $P = (X - 2)^3(X + 1)$ — which one checks by expanding $(X-2)^3 = X^3 - 6X^2 + 12X - 8$ and multiplying by $X + 1$. The insight: multiplicities are read off evaluations, no factorization needed — which is exactly how one detects them when the factorization is out of reach.

Example 8.13 (Detecting multiple roots with a gcd)

When no root is known, Proposition 8.11 still gives a global multiple-root detector: $a$ is a multiple root of $P$ iff it is a common root of $P$ and $P'$, so $P$ has a multiple root (in $\C$) iff $\gcd(P, P') \neq 1$ — computable by the Euclidean algorithm without solving anything. Sample: $P = X^3 - 3X + 2$, $P' = 3X^2 - 3 = 3(X - 1)(X + 1)$. Testing the roots $\pm1$ of $P'$ inside $P$: $P(1) = 0$ but $P(-1) = 4$, so

the root $1$ is multiple; dividing twice, $P = (X - 1)^2(X + 2)$. The gcd even reports the full set of multiple roots, each with multiplicity lowered by one — the fact every computer-algebra system exploits to “square-free factorize” before any root-hunting, and the polynomial twin of Exercise 8.9’s no-multiple-root arguments.

Example 8.17

$X^4 + 4$ was factored over $\R$ in Exercise 3.5 by pairing the four complex roots $\pm 1 \pm \iu$: $X^4 + 4 = (X^2 - 2X + 2)(X^2 + 2X + 2)$. Neither quadratic splits over $\R$ (discriminants $-4$). Note: an irreducible real polynomial has degree $1$ or $2$ — this is exactly what the factorization theorem says. The same conjugate-pairing run on $X^4 + 1$, whose roots are $\eu^{\pm\iu\pi/4}$ and $\eu^{\pm3\iu\pi/4}$: each pair contributes $X^2 - 2\cos\theta\,X + 1$, so

an identity invisible to naive factoring attempts over $\Q$ — the price of insisting on real (here even irrational) coefficients, and a standard input for integrating $\frac1{x^4 + 1}$ in Chapter 15.