Rational Zeros Calculator

Find all possible and actual rational zeros of polynomials using the Rational Root Theorem

Polynomial Setup

Your Polynomial:
P(x) = x^3 + 1

Note: All coefficients must be integers. If your polynomial has fractional coefficients, multiply through to clear denominators.

Rational Zeros Analysis

Leading Coefficient:
1
Constant Term:
1
Factors of Leading Coefficient:
±1
Factors of Constant Term:
±1
All Possible Rational Zeros (2 total):
-1, 1
Actual Rational Zeros (1 found):
-1

Summary

  • • By the Rational Root Theorem: ±(factor of constant)/(factor of leading coefficient)
  • • Found 2 possible rational zeros
  • • Verified 1 actual rational zeros

Verification

P(-1) = 0

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

Theorem Statement:
If P(x) has integer coefficients and p/q is a rational root (in lowest terms), then p divides the constant term and q divides the leading coefficient.
Formula: Rational zeros = ±(factors of a₀)/(factors of aₙ)

Solution Steps

1
Find factors of the constant term (a₀)
2
Find factors of the leading coefficient (aₙ)
3
Form all possible fractions ±p/q
4
Simplify fractions and remove duplicates
5
Test each candidate using substitution or synthetic division

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:

  1. Direct substitution into the polynomial
  2. Using synthetic division (more efficient)
  3. 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!