To factorize the expression x^3−2x^2+3x−6, we first identify a value of x for which the value of function is 0. This could be a factor of last term i.e. -6 i.e. among (1, 2, 3, 6, -1, -2, -3, -6).
It is seen that for x-2 function is zero. Hence (x-2) is a factor of x^3−2x^2+3x−6. Dividing latter by former, we get the factors of the function as
(x-2)(x^2+3)
Now as the determinant (b^2-4ac if the function is ax^2+bx+c) of x^2+3 is
0^2-4.1.3 = -12. a negative number, this cannot be factorized into rational factors (assuming that to be a condition).
The factors are hence (x-2)(x^2+3)
