Rational Root Theorem Calculator – Find Polynomial Roots
Find possible rational roots of polynomials
Table of Contents
How to Use
- Enter polynomial coefficients from highest to lowest degree
- Separate coefficients with commas or spaces
- Click calculate to see all possible rational roots
- Test each possible root to find actual zeros
What is the Rational Root Theorem?
The Rational Root Theorem (also called the Rational Zero Theorem) provides a way to find all possible rational roots of a polynomial equation with integer coefficients. It states that if p/q is a rational root of the polynomial, then p must be a factor of the constant term and q must be a factor of the leading coefficient.
Formula: For polynomial aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ = 0, possible rational roots are ±p/q where p divides a₀ and q divides aₙ.
How to Apply the Rational Root Theorem
- Identify the leading coefficient (aₙ) and constant term (a₀)
- List all factors of the constant term (these are possible values of p)
- List all factors of the leading coefficient (these are possible values of q)
- Form all possible fractions ±p/q
- Test each possible root by substituting into the polynomial
Example
For the polynomial 2x³ - 3x² - 8x + 12 = 0:
- Leading coefficient (aₙ) = 2, factors: 1, 2
- Constant term (a₀) = 12, factors: 1, 2, 3, 4, 6, 12
- Possible rational roots: ±1, ±2, ±3, ±4, ±6, ±12, ±1/2, ±3/2
Testing these values reveals that x = 2, x = 3/2, and x = -2 are the actual roots.
Important Notes
- The theorem only finds possible rational roots, not irrational or complex roots
- Not all possible roots listed will be actual roots
- Each possible root must be tested by substitution
- The polynomial must have integer coefficients for the theorem to apply
Frequently Asked Questions
- What if none of the possible rational roots work?
- If none of the possible rational roots are actual roots, the polynomial may have only irrational or complex roots. You would need to use other methods like the quadratic formula, synthetic division, or numerical methods to find these roots.
- Can the Rational Root Theorem find all roots of a polynomial?
- No, the theorem only helps find rational roots. Polynomials can have irrational roots (like √2) or complex roots (like 2+3i) that this theorem cannot identify. It's a starting point for finding roots, not a complete solution.
- Why do we use ±p/q instead of just p/q?
- We use both positive and negative values because a polynomial can have both positive and negative roots. For example, x² - 4 = 0 has roots x = 2 and x = -2. Using ±p/q ensures we don't miss any potential roots.
- Does the order of coefficients matter?
- Yes! Coefficients must be entered from highest degree to lowest degree (constant term last). For example, for 3x² + 2x - 5, enter: 3, 2, -5. Missing terms should be entered as 0.