Rational Zeros Calculator
Find all possible and actual rational zeros of polynomials using the Rational Root Theorem
Polynomial Setup
Note: All coefficients must be integers. If your polynomial has fractional coefficients, multiply through to clear denominators.
Rational Zeros Analysis
Summary
- • By the Rational Root Theorem: ±(factor of constant)/(factor of leading coefficient)
- • Found 2 possible rational zeros
- • Verified 1 actual rational zeros
Verification
Example Calculation
Example: P(x) = 2x³ - 3x² - 8x + 12
Leading coefficient: 2, factors: ±1, ±2
Constant term: 12, factors: ±1, ±2, ±3, ±4, ±6, ±12
Possible rational zeros: ±1, ±2, ±3, ±4, ±6, ±12, ±1/2, ±3/2
Testing Process
1. Test each possible zero by substitution or synthetic division
2. P(2) = 2(8) - 3(4) - 8(2) + 12 = 16 - 12 - 16 + 12 = 0 ✓
3. Continue testing to find all rational zeros
Rational Root Theorem
Solution Steps
Important Notes
All coefficients must be integers
The theorem only gives possible zeros, not guaranteed zeros
A polynomial may have no rational zeros
Use synthetic division to verify candidates
Understanding the Rational Root Theorem
What is the Rational Root Theorem?
The Rational Root Theorem is a powerful tool in algebra that helps us find all possible rational zeros (roots) of a polynomial with integer coefficients. It states that any rational zero p/q (in lowest terms) of a polynomial must have p as a factor of the constant term and q as a factor of the leading coefficient.
Why is it Important?
- •Provides a systematic way to find rational zeros
- •Limits the search to a finite set of candidates
- •Essential for factoring higher-degree polynomials
- •Foundation for solving polynomial equations
Mathematical Foundation
For polynomial P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀
Possible rational zeros = ±(factors of a₀)/(factors of aₙ)
Testing Candidates
Once you have the list of possible rational zeros, you need to test each one by:
- Direct substitution into the polynomial
- Using synthetic division (more efficient)
- Checking if the remainder is zero
Remember: The theorem only tells you the possible rational zeros. A polynomial might have no rational zeros at all!