📐 Rational Zeros Calculator
Find possible rational zeros using Rational Root Theorem
For polynomial: ax^n + ... + c
How to Use This Calculator
Enter Leading Coefficient
Type the coefficient of the highest-degree term (a). For 2x³ + 5x + 6, enter 2.
Enter Constant Term
Type the constant term (c). For 2x³ + 5x + 6, enter 6.
Click Find
See all possible rational zeros using the Rational Root Theorem.
Formula
Possible Rational Zeros = ± (factors of c) / (factors of a)
For polynomial: a_nx^n + ... + a₁x + a₀
Example 1: x³ - 2x² - 5x + 6 = 0
Leading coefficient (a) = 1, Constant (c) = 6
Factors of 6: ±1, ±2, ±3, ±6
Factors of 1: ±1
Possible zeros: ±1/1, ±2/1, ±3/1, ±6/1
Possible zeros: 1, -1, 2, -2, 3, -3, 6, -6
Example 2: 2x³ + 5x + 1 = 0
a = 2, c = 1
Possible zeros: ±1/1, ±1/2
Possible zeros: 1, -1, 1/2, -1/2
About Rational Zeros Calculator
The Rational Zeros Calculator uses the Rational Root Theorem to find all possible rational zeros of a polynomial. This theorem states that any rational zero of a polynomial must be of the form p/q, where p divides the constant term and q divides the leading coefficient.
When to Use This Calculator
- Finding Roots: Get all candidate rational roots
- Factoring: Start factoring process
- Polynomials: Solve higher-degree equations
- Algebra: Polynomial division and factoring
Why Use Our Calculator?
- ✅ Complete List: Shows all possible rational zeros
- ✅ Fast Results: Instant calculation
- ✅ Educational: Understand Rational Root Theorem
- ✅ Free Tool: No registration
Key Points
- Not all possible zeros are actual zeros
- Test each candidate by substitution
- Works only for polynomials with integer coefficients
- May include duplicates and need simplification
Tips
- Start with simplest candidates (usually ±1)
- Use synthetic division to test zeros
- Even irrational zeros won't appear in this list
Frequently Asked Questions
What is the Rational Root Theorem?
A theorem stating that any rational zero must be p/q where p divides the constant term and q divides the leading coefficient.
Are all listed values actual zeros?
No! This gives possible candidates. You must test each one to see if it's actually a zero.
What about irrational zeros?
This calculator only finds rational candidates. Irrational zeros like √2 won't appear in the list.