Question

How do you use the rational root theorem to list all possible roots for `12x^4+14x^3-5x^2-14x-4=0`?

Answer 1

The only possible rational roots are:

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

...but none of these is a root.

So this equation has no rational roots.

Explanation:

Given:

`12x^4+14x^3-5x^2-14x-4 = 0`

By the rational root theorem, any rational roots must be expressible in the form `p/q` for integers `p, q` with `p` a divisor of the constant term `-4` and `q` a divisor of the coefficient `12` of the leading term.

So the only possible rational roots are:

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

In practice, none of these is a root, so this quartic equation has no rational roots.