Rational Root Theorem
2026-02-28 13:12 Diff
https://brightchamps.com/en-us/math/algebra/rational-root-theorem

According to the rational root theorem, for the root of a polynomial to be a rational number, the numerator and denominator must be factors of the constant term and leading coefficient, respectively. The leading coefficient is the coefficient of the term that has the highest power of the variable.

For a polynomial:

\(P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 \),

\(a_n\) denotes the leading coefficient and a0 denotes the constant term. 

Rational Root Theorem Statement

The rational root theorem states: 
 

If a polynomial equation \(p(x) = anxn + an-1xn-1 + . . . + a1x +a0\), has a rational root \(\frac{p}{q} \) in its lowest terms, then:
 

  • p must be a factor (divisor) of the constant term a₀, and
     
  • q must be a factor (divisor) of the leading coefficient, aₙ.
     

Any possible rational solution to the polynomial must be formed by dividing a factor of the constant term by a factor of the leading coefficient.

Therefore, possible rational roots of \(\frac{p}{q} \) are of the form \(\frac{\text{factors of } a_0}{\text{factors of } a_n} \).

Rational Root Theorem Definition


The rational root theorem is a mathematical rule for finding all possible rational zeroes (or roots) of a polynomial equation with integer coefficients. It states that if a polynomial has a rational root written in the form \(\frac{p}{q}\) in its lowest terms, then the numerator p must be a factor of the constant term, and the denominator q must be a factor of the leading coefficient.


Rational Root Theorem Examples
 

Example 1: Find the possible rational roots of \(p(x) = x^3-6x^2 + 11x - 6\). 
Here, the constant term a0 = -6.
Leading coefficient an = 1. 
Factors of -6 are ±1, ±2, ±3, ±6. 
Factors of 1 are ±1. 
The possible rational roots are ±1, ±2, ±3, and ±6. 

Example 2: Find the possible rational roots of \(p(x) = 2x^3 + x^2 - 7x - 6\). 
Here, constant term a0 = -6. 
Leading coefficient an = 2. 
Factors of -6 are ±1, 2, ±3, and ±6. 
Factors of 2 are ±1 and ±2. 
Therefore, the possible rational roots are ±1, ±2, ±3, ±6, ±1/2, ±3/2