Rational Root Theorem Calculator
Find all possible rational roots of a polynomial using the Rational Root Theorem.
Find all possible rational roots: ±p/q where p|constant, q|leading coeff
What Is the Rational Root Theorem Calculator?
The Rational Root Theorem Calculator lists all possible rational roots of a polynomial with integer coefficients. Enter the polynomial coefficients and the tool generates every candidate rational root p/q where p divides the constant term and q divides the leading coefficient.
Formula
How to Use
Enter the coefficients of your polynomial from highest to lowest degree. The calculator extracts the constant term and leading coefficient, finds all their factors, and lists every possible rational root in the form ±p/q. You can then test each candidate using synthetic division.
Example Calculation
Polynomial: 2x³ − x² − 5x + 3. Leading coeff = 2, factors: {1,2}. Constant term = 3, factors: {1,3}. Possible rational roots: ±1/1, ±1/2, ±3/1, ±3/2 = {±1, ±½, ±3, ±3/2}. Testing x=3/2: 2(27/8)−(9/4)−15/2+3 = 27/4−9/4−30/4+12/4 = 0 ✓
Understanding Rational Root Theorem
The Rational Root Theorem is a powerful algebraic tool that narrows down the possible rational roots of any polynomial with integer coefficients to a finite list. Rather than trying an infinite set of possibilities, you test at most p × q candidate values, where p and q are the number of factors of the constant and leading terms respectively.
Once you have the candidate list, synthetic division is the fastest way to test each one. If a candidate c is a root, synthetic division gives zero remainder and produces a polynomial of degree one less — which may reveal further rational roots or simpler factors.
This theorem is most useful for polynomials of degree 3 and above, where no general closed-form solution exists (unlike quadratics). Combined with the Factor Theorem and synthetic division, the Rational Root Theorem provides a systematic strategy for fully factoring polynomials with rational coefficients.
Frequently Asked Questions
What is the Rational Root Theorem?
If a polynomial with integer coefficients has a rational root p/q (in lowest terms), then p must divide the constant term and q must divide the leading coefficient.
Does the theorem guarantee rational roots exist?
No. The theorem only lists candidates. Many polynomials have no rational roots at all, only irrational or complex roots.
How do I test a candidate root?
Substitute the candidate into the polynomial. If it evaluates to zero, it is a root. Alternatively, use synthetic division — if the remainder is zero, the candidate is a root and the quotient is the deflated polynomial.
What do I do after finding a rational root?
Use synthetic division to factor out (x − root) from the polynomial, reducing its degree by one. Repeat the process on the quotient until it is fully factored.
Is this calculator free?
Yes, completely free with no sign-up required.
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