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

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Answer 1 · 2015-05-30T18:52:19

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.