Question

How do you find all the rational zeros of a polynomial function?

Answer 1

You can use the rational root theorem:

Given a polynomial of the form:

`a_0x^n+a_1x^(n-1)+...+a_n` with `a_0,...,a_n` integers,

all rational roots of the form `p/q` written in lowest terms (i.e. with `p` and `q` having no common factor) will satisfy.

`p | a_n` and `q | a_0`

That is `p` is a divisor of the constant term and `q` is a divisor of the coefficient of the highest order term.

This gives you a finite number of possible rational roots to try.

For example, the rational roots of

`6x^4-7x^3+x^2-7x-5=0`

must be of the form `p/q` where `p` is `+-1` or `+-5` and
`q` is `1`, `2`, `3` or `6`.

You can try substituting each of the possible combinations of `p` and `q` as `x=p/q` into the polynomial to see if they work.

In fact the only rational roots it has are `-1/2` and `5/3`.

Once you have found one root, you can divide the polynomial by the corresponding factor to simplify the problem.