Rule:
1. Make sure the polynomial is in descending order. (This one is)
Write down all the integers, positive and negative which can be divided
evenly in the LAST term or coefficient.
{This one has last term 8, and its factors are the positive and negative
integers ±1, ±2, ±4, and ±8.
2. Write down all the positive integers which can be divided
evenly in the coefficient of the first term.
(This one has first term x4 which has coefficient 1. 1 has only
the factor 1),
3. Make all possible fractions using those from step 1 as numerators, and
all those from step 2 as denominators:
All possible rational roots are:
,
, 
, 
Since the denominators are all 1, the rational roots are
±1, ±2, ±4, and ±8.
When the first term has coefficient 1, as this one does, you need only go to
step 2.
[However, as it turns out, this polynomial equation has NO rational roots.
So in the strict sense of the word "possible", no rational roots are
possible. However algebra textbooks always use "possible" in the loose sense,
of "maybe and maybe not".]
Edwin
