Which of the following is the factor of the equation $x^3-3x^2+5x-6$?

Let's examine all the options given above.

• When $(x+2)$ i.e. $x=-2$

$x^3-3x^2+5x-6$

Let's replace x with $-2$.

$$\begin{gathered} {(-2)}^3-3{(-2)}^2+5(-2)-6 \\ = -8-12-10-6 \\ = -36 \end{gathered}$$

Therefore it is not the correct answer.

• When $(x-3)$ i.e. $x=3$

$x^3-3x^2+5x-6$

Let's replace x with $3$.

$$\begin{gathered} =3^3-3{\left(3\right)}^2+5\left(3\right)-6 \\ =27-27+15-6 \\ = 0+9 \\ = 9 \end{gathered}$$

Therefore it is not the correct answer.

• When $(x-2)$ i.e. $x=2$

$x^3-3x^2+5x-6$

Let's replace x with $2$.

$$\begin{gathered} {\left(2\right)}^3-3{\left(2\right)}^2+5\left(2\right)-6 \\ = 8-12+10-6 \\ = -4+10-6 \\ = 6-6 \\ = 0 \end{gathered}$$

Therefore, $(x-2)$ is the correct answer.