High School Polynomial Root Finder
Enter coefficients separated by spaces (highest degree first).
Example: 1 -6 11 -6 for x³ – 6x² + 11x – 6
Rational Root Candidates (p/q)
Found Rational Roots
Testing Steps
Polynomial Graph
Understanding the Polynomial Root Finder
Finding the values of x where a polynomial equals zero is one of the fundamental problems in algebra. For quadratic equations (degree 2), we have a simple formula. However, for cubic (degree 3), quartic (degree 4), and higher-degree polynomials, solving for roots becomes significantly more complex. This Polynomial Root Finder utilizes the Rational Root Theorem to identify potential rational solutions systematically.
The Rational Root Theorem
This theorem states that if a polynomial has integer coefficients, every rational zero of the function f(x) has the form p/q, where:
- p is a factor of the constant term (the last number).
- q is a factor of the leading coefficient (the first number).
For example, if f(x) = 2x³ + … + 3, then p must be a factor of 3 (±1, ±3) and q must be a factor of 2 (±1, ±2). The possible rational roots are ±1, ±3, ±1/2, ±3/2.
How to use the Tool
Simply enter the coefficients of your polynomial in descending order of degree. For example, for x³ – 7x – 6, you would enter “1 0 -7 -6” (notice the 0 for the missing x² term). The tool will calculate all candidates, test them, and graph the function to visually confirm the x-intercepts.
Why use this Polynomial Root Finder?
Manual substitution using Synthetic Division or the Remainder Theorem can be tedious when there are many candidates. This tool automates the “Guess and Check” phase of solving high-degree polynomials, allowing students to focus on the logic of factoring rather than the arithmetic.
FAQ
Yes, this tool is free, open-source, and runs entirely in your browser. No installation is required.
The tool will indicate “None found.” The roots might be irrational (like √2) or complex numbers, which the Rational Root Theorem cannot find directly.
The Rational Root Theorem section only finds rational numbers. However, the graph visualizes real roots. Complex roots do not touch the X-axis.
Yes! If your equation is x³ – 1, you must enter “1 0 0 -1” to account for the missing x² and x terms.