1. What is the Rational Roots Theorem?

The Rational Roots Theorem (or Rational Zeros Theorem) provides a complete list of possible rational roots of a polynomial equation with integer coefficients. It states that any possible rational root, expressed in lowest terms p/q, has p as a factor of the constant term and q as a factor of the leading coefficient.

2. How Does the Calculator Work?

The calculator uses the Rational Roots Theorem formula:

Where:

• \( p \) — All integer factors of the constant term
• \( q \) — All integer factors of the leading coefficient

Explanation: The calculator generates all possible combinations of p/q (both positive and negative) and removes duplicates.

3. Importance of Rational Roots

Details: Finding rational roots is often the first step in solving polynomial equations. It helps factor polynomials and find all real roots, both rational and irrational.

4. Using the Calculator

Tips: Enter comma-separated lists of factors for both the constant term (p) and leading coefficient (q). For example, if p=6 and q=2, enter "1,2,3,6" for p and "1,2" for q.

5. Frequently Asked Questions (FAQ)

Q1: Does this guarantee all roots are rational?
A: No, it only lists possible rational roots. The actual roots might be irrational or complex.

Q2: What if my polynomial has no constant term?
A: If the constant term is zero, factor out x first and apply the theorem to the remaining polynomial.

Q3: How do I test which roots are actual roots?
A: Use synthetic division or plug each possible root into the polynomial to see if it yields zero.

Q4: What about repeated roots?
A: The calculator shows each possible root only once, but actual roots may have multiplicity.

Q5: Can this be used for polynomials with non-integer coefficients?
A: No, the theorem only applies to polynomials with integer coefficients.