Question

How do you find the roots of `x^3-2x^2-5x+6=0`?

Answer 1

`x in {+1, +3, -2}`

Explanation:

If we note that the sum of the coefficients of the terms equals zero
then it follows that `x=1` is one solution to this equation.

That is `(x-1)` is a factor of the right side.
Either using synthetic or long division we can obtain
`color(white)("XXX")(x^3-2x^2-5x+6)div(x-1)=color(green)(x^2-x-6)`

Using basic principles we can factor
`color(white)("XXX")(x^2-x-6) = (x-3)(x+2)`

So `x^3-2x^2-5x+6=0`

`rarr (x-1)(x-3)(+2)=0`

giving the solutions
`color(white)("XXX")x=+1 or x=+3 or x=-2`