The rational zero theorem states that if the coefficients of the polynomial
 |
(1)
|
are integers with 
, then every nonzero rational
root 
,
written with relatively prime integers 
and 
, has a numerator 
that divides 
and a denominator 
that divides 
. This follows since a polynomial
of polynomial order 
with 
rational roots can be expressed,
with integer coefficients
throughout, as
 |
(2)
|
where the rational roots are 
, 
, ..., and 
, each written in lowest terms. Factoring out the
s,
 |
(3)
|
Now, multiplying through,
 |
(4)
|
where we have not bothered with the other terms. Since the first and last coefficients are 
and 
,
all the nonzero rational roots are ratios of divisors of 
to divisors of 
.
If 
,
this condition on divisors alone does not give a finite
list of possible nonzero roots. First factor
out the largest power 
dividing the polynomial.
Then 0 is a root of multiplicity 
. The usual finite rational
root test applies to the remaining polynomial,
whose value at 0 is nonzero.
