Multiplicity of a root

Compute derivatives.
$$P(X)=(X-1)^4$$ (by recognizing the binomial expansion, or computing). Let us verify by successive derivatives:
- $P(1)=1-4+6-4+1=0$.
- $P'(X)=4X^3-12X^2+12X-4$, $P'(1)=4-12+12-4=0$.
- $P''(X)=12X^2-24X+12$, $P''(1)=12-24+12=0$.
- $P'''(X)=24X-24$, $P'''(1)=0$.
- $P^{(4)}(X)=24$, $P^{(4)}(1)=24\neq0$.

Conclusion. The multiplicity of $1$ is $4$, so $P(X)=(X-1)^4$.