Question

How do you find all rational roots for `x^3 - 3x^2 + 4x - 12 = 0`?

Answer 1

The only rational root of `x^3-3x^2+4x-12=0` is `3`.

Explanation:

`x^3-3x^2+4x-12=0` can have one root among factors of `12` i.e. `{1,-1,2,-2,3,-3,4,-4,6,-6,12,-12}`, if at least one root is rational.

It is apparent that `3` satisfies the equation, hence `x-3` is a factor of `x^3-3x^2+4x-12`. Dividing latter by `(x-3)`, we get

`x^3-3x^2+4x-12=x^2(x-3)+4(x-3)=(x^2+4)(x-3)`

`x^2+4=0` does not have rational rots as discriminant `b^2-4ac=0-4*1*4=-16`

hence the only rational root of `x^3-3x^2+4x-12=0` is `3`.

The two roots will be imaginary numbers `-2i` and `+2i`.