Question

How do you use the rational roots theorem to find all possible zeros of `3x^5-7x^2+x+6`?

Answer 1

This quintic has no rational zeros.

Explanation:

`f(x) = 3x^5-7x^2+x+6`

By the rational root theorem, any rational zeros of `f(x)` are expressible in the form `p/q` for integers `p, q` with `p` a divisor of the constant term `6` and `q` a divisor of the coefficient `3` of the leading term.

That means that the only possible rational zeros are:

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

Evaluating `f(x)` for each of these, we find none work.

So this quintic has no rational zeros.

In fact it has one Real zero at `x~~ -0.873` and two pairs of Complex zeros.

These zeros are not expressible in terms of `n`th roots, so about the best you can do is find numerical approximations using Newton's method or the Durand-Kerner method.