Factorization of a cubic

#cubic#factorization#roots

Statement

Factor $P(X)=X^3-3X^2+4$ over $\mathbb{R}$ .

Hint

Test small integers for roots.

Show solutionHide solution

Test. $P(2)=8-12+4=0$ . Root. $P(-1)=-1-3+4=0$ . Root.

Factor. Dividing by $(X-2)(X+1)=X^2-X-2$ : $P=(X-2)(X+1)(X-2)=(X-2)^2(X+1)$ .

Conclusion. $P(X)=(X-2)^2(X+1)$ . Roots: $2$ (double), $-1$ (simple).