Rational Root Calculator Polynomial

What is the Rational Root Theorem?

The Rational Root Theorem states that any possible rational root, expressed in lowest terms p/q, of a polynomial equation with integer coefficients must satisfy two conditions:

1. The integer p must be a factor of the constant term (the term without a variable).
2. The integer q must be a factor of the leading coefficient (the coefficient of the highest power of the variable).

This theorem helps limit the number of possible rational roots you need to test when solving polynomial equations.

How to Use the Calculator

1. Enter the coefficients of your polynomial in the input fields. For example, for 2x³ - 3x² + 4x - 6, you would enter 2 for the x³ coefficient, -3 for the x² coefficient, 4 for the x coefficient, and -6 for the constant term.
2. Click the "Calculate" button to find all possible rational roots based on the Rational Root Theorem.
3. Review the results to determine which of the possible roots are actual roots of your polynomial equation.

Example Calculation

Let's find the possible rational roots for the polynomial 2x³ - 3x² + 4x - 6.

Following the Rational Root Theorem:

1. Factors of the constant term (-6): ±1, ±2, ±3, ±6
2. Factors of the leading coefficient (2): ±1, ±2

The possible rational roots are all combinations of these factors:

• ±1, ±1/2
• ±2, ±2/2 (which simplifies to ±1)
• ±3, ±3/2
• ±6, ±6/2 (which simplifies to ±3)

After removing duplicates and simplifying, the possible rational roots are: ±1, ±1/2, ±3, ±3/2.

Limitations

The Rational Root Theorem only provides possible rational roots. It does not guarantee that all these roots are actual roots of the polynomial equation. You must test each possible root to determine if it is an actual root.

Additionally, the theorem only applies to polynomials with integer coefficients. If your polynomial has fractional coefficients, the theorem does not apply directly.