Question

How do you use the rational zero theorem to list all possible rational zeros for the given function `f(x)=x^3-14x^2+13x-14`?

Answer 1

Hello,

Let `P = a_n X^n + a_{n-1}X^{n-1} + \ldots + a_1X+a_0` a polynom with `a_0,\ldots, a_n` integers. Suppose that `a_n\ne 0` and `a_0\ne 0`.

If irreductible fraction `\frac{p}{q}` is a root of `P`, then `p` is a factor of `a_0` and `q` is a factor of `a_n`.

In your example, the factors of `a_0=-14` are
`-1,1,-2,2-7,7,-14,14`,
and the factors of `a_n=1` are just `-1,1`.

Therefore, you have some candidates for the rational roots :
`-1,1,-2,2-7,7,-14,14`.

Remark. If `a_n=1`, the rational roots are necessarily integers !

You can check that no one is solution !