Polynomial Root Theorems
Rule

Rational Root Theorem

Consider a polynomial where every coefficient is an integer.
The following properties are true for the roots of the polynomial.
  1. Every integer root is a factor of the constant
  2. All rational roots must have a numerator that is a factor of and a denominator that is a factor of the leading coefficient

Proof

Rational Root Theorem
To prove this theorem, each of the properties will be considered individually. To start with the first property, consider a polynomial with integer coefficients that has an integer root This means that
Now is subtracted from both sides of the equation.
Since the coefficients of are integers and is an integer, the expression between the parentheses results in an integer. Also, is the product of an integer and which means that is a factor of This completes the proof of the first property.
-a0 = x_r * k

For the second property, suppose that the polynomial has a rational root such that the fraction is written in its simplest form. Again, it is obtained that

a_n \left(\dfrac{p}{q}\right)^n + a_{n-1}\left(\dfrac{p}{q}\right)^{n-1}+\cdots+a_1\left(\dfrac{p}{q}\right)+a_0 = 0
This time the equation will be modified in a different way.
Since the coefficients of and are all integers, the expression between parentheses results in an integer. Also, since is written in its simplest form, and do not have a common factor, which means that and do not have a common factor either. Therefore, is a factor of

The numerator of the root, is a factor of

If the modification is done differently, it is possible to come to the other property.
With the same reasoning as before, it is possible to conclude that the denominator of the root, is a factor of finishing the proof.
Exercises