Rational Zeros Calculator

Rational Zeros Calculator

The Rational Zeros Calculator helps find all possible rational zeros (roots) of a polynomial equation using the Rational Root Theorem. This theorem states that if a polynomial has a rational zero, it must be in the form of \( \frac{p}{q} \), where \( p \) is a factor of the constant term, and \( q \) is a factor of the leading coefficient.

Historical Background

The Rational Root Theorem is a result in algebra that provides a criterion for the possible rational roots of a polynomial equation. It is a valuable tool for narrowing down the search for the exact roots of polynomials and has been used in algebra since it was formalized in the 17th century.

Calculation Process

1. List the factors of the constant term (the last coefficient).
2. List the factors of the leading coefficient (the first coefficient).
3. Form all possible fractions \( \frac{p}{q} \), where \( p \) is a factor of the constant term, and \( q \) is a factor of the leading coefficient.
4. Simplify the fractions and list them as possible rational zeros.

Example Calculation

For the polynomial \( 2x^3 - 3x^2 + x - 6 \):

1. Constant term: -6. Factors: ±1, ±2, ±3, ±6.
2. Leading coefficient: 2. Factors: ±1, ±2.
3. Possible zeros: \( \frac{1}{1}, \frac{1}{2}, \frac{2}{1}, \frac{2}{2}, \dots \).
4. Simplified possible rational zeros: ±1, ±1/2, ±2, ±3, ±3/2, ±6.

This calculator efficiently determines all possible rational zeros, aiding in polynomial factorization and root finding.