Rational Zeros Calculator
Find all possible and actual rational zeros of a polynomial using the Rational Root Theorem
Enter coefficients in order: aₙ, aₙ₋₁, ..., a₁, a₀
Polynomial P(x):
Actual Rational Zeros
x =
Found rational zero(s)
No Rational Zeros
The polynomial may have irrational or complex zeros
Factors of Constant Term (p)
a₀ =
Factors:
Factors of Leading Coefficient (q)
aₙ =
Factors:
All Possible Rational Zeros (p/q)
Highlighted values are actual zeros
Step-by-Step Solution
Rational Zeros Theorem
For a polynomial P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ with integer coefficients:
Possible Rational Zero = ±p/q
where p divides a₀ and q divides aₙ
- p = any factor of the constant term (a₀)
- q = any positive factor of the leading coefficient (aₙ)
- Include both positive and negative values of p
- Test each candidate by substituting into P(x)
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About Rational Zeros Calculator
What is a Rational Zeros Calculator?
A rational zeros calculator is a mathematical tool that finds all possible rational roots (zeros) of a polynomial equation with integer coefficients. It uses the Rational Root Theorem (also called the Rational Zeros Theorem) to systematically identify which values could be roots of the polynomial.
The Rational Zeros Theorem
If a polynomial P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ has integer coefficients, then any rational zero p/q (in lowest terms) must satisfy:
- p is a factor of the constant term a₀
- q is a factor of the leading coefficient aₙ
Example
For the polynomial P(x) = 2x³ - 3x² - 8x + 12:
- Constant term a₀ = 12, factors: ±1, ±2, ±3, ±4, ±6, ±12
- Leading coefficient aₙ = 2, factors: ±1, ±2
- Possible rational zeros (p/q): ±1, ±2, ±3, ±4, ±6, ±12, ±1/2, ±3/2
How to Use This Calculator
- Enter polynomial coefficients in order from highest degree to constant term
- View all possible rational zeros generated by the p/q formula
- See actual rational zeros that make the polynomial equal to zero
- Review the step-by-step solution showing the factor analysis
Algorithm Steps
- Identify the constant term (a₀) and leading coefficient (aₙ)
- List all factors of a₀ (these are the p values)
- List all factors of aₙ (these are the q values)
- Form all possible fractions p/q (including ± signs)
- Remove duplicates and reduce fractions
- Test each candidate by substituting into the polynomial
- Those that evaluate to zero are actual rational zeros
Important Notes
- This theorem only finds rational zeros; polynomials may also have irrational or complex zeros
- Not all possible rational zeros are actual zeros—each must be tested
- If the leading coefficient is 1 (monic polynomial), possible zeros are just the factors of the constant term
Applications
- Factoring polynomials: Finding rational roots helps factor polynomials
- Solving equations: Identify starting points for finding exact solutions
- Algebra courses: Essential technique in precalculus and algebra
- Engineering: Analyzing system equations and transfer functions
Tip: Once you find a rational zero, you can use synthetic division to reduce the polynomial degree and find remaining zeros more easily.