Question

How do you find all rational roots for `x^5 - 2x^4 + 11x^3 - 22x^2 - 12x + 24 = 0`?

Answer 1

`+-1,2`

Explanation:

We will use the Rational Root Theorem:

If the rational number r/s is a root of a polynomial whose coefficients are integers, then the integer r is a factor of the constant term, and the integer s is a factor of the leading coefficient.

r=24 and s=1

So the rational roots must be factor of 24/1=24:

`+-1, +-2,+-3,+-4,+-6,+-8, +-12`

By trying these possible roots, we discover that `+-1,2` are roots of the polynom.

If we divide the polynom by `(x-1)(x+1)(x-2)` we will obtain `x^2+12` which has the roots `+-isqrt(12)`, which are not rational.