The factor theorem

#factor theorem#roots#divisibility

Statement

Show that $X-2$ divides $P(X)=X^3-4X^2+X+6$ , then factor $P$ completely.

Hint

Compute $P(2)$ .

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Root test. $P(2)=8-16+2+6=0$ , so $2$ is a root. By the factor theorem, $X-2\mid P$ .

Division.
$P=(X-2)(X^2-2X-3)=(X-2)(X-3)(X+1).$

Conclusion. $P(X)=(X-2)(X-3)(X+1)$ . Roots: $2,3,-1$ .