1. What Is The Rational Zero Theorem?

The Rational Zero Theorem provides a complete list of possible rational zeros (roots) of a polynomial function with integer coefficients. It states that any possible rational zero of a polynomial is a ratio p/q where p is a factor of the constant term and q is a factor of the leading coefficient.

2. How Does The Calculator Work?

The calculator uses the Rational Zero Theorem formula:

Where:

• Constant term — The term without any variable (a₀)
• Leading coefficient — The coefficient of the highest degree term (aₙ)

Explanation: The calculator finds all factors of the constant term and leading coefficient, then generates all possible combinations of p/q.

3. Importance Of Finding Rational Zeros

Details: Finding rational zeros helps in factoring polynomials completely, solving polynomial equations, and analyzing polynomial functions. It's a fundamental step in polynomial analysis.

4. Using The Calculator

Tips: Enter the polynomial coefficients from highest to lowest degree, separated by commas. For example, for 2x³ - 3x² - 11x + 6, enter "2,-3,-11,6".

5. Frequently Asked Questions (FAQ)

Q1: Does this guarantee all zeros are rational?
A: No, it only lists possible rational zeros. The polynomial might have irrational or complex zeros.

Q2: What if my polynomial has non-integer coefficients?
A: Multiply through by the least common denominator to convert to integer coefficients first.

Q3: How do I test which zeros actually work?
A: Use synthetic division or direct substitution to verify which candidates are actual zeros.

Q4: What's the difference between zeros and roots?
A: They're often used interchangeably, but technically zeros refer to function values (f(x)=0) while roots refer to equation solutions.

Q5: Can this be used for higher degree polynomials?
A: Yes, the theorem applies to any degree polynomial with integer coefficients.