1. What is the Rational Zero Theorem?

The Rational Zero Theorem states that any possible rational root of a polynomial equation with integer coefficients is of the form ±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:

Steps:

1. Finds all factors of the constant term (p)
2. Finds all factors of the leading coefficient (q)
3. Creates all possible combinations of ±p/q
4. Simplifies fractions to lowest terms
5. Removes duplicates
6. Sorts the results numerically

3. Importance of Rational Zero Theorem

Details: The theorem provides a systematic way to find all possible rational roots of a polynomial equation, which is the first step in solving many polynomial equations.

4. Using the Calculator

Tips: Enter the constant term and leading coefficient of your polynomial. The calculator will show all possible rational roots. Remember these are only possibilities - not all may be actual roots.

5. Frequently Asked Questions (FAQ)

Q1: What if my polynomial has non-integer coefficients?
A: The Rational Zero Theorem only applies to polynomials with integer coefficients. Multiply through by the least common denominator to convert to integer coefficients first.

Q2: What if there are no rational roots?
A: The polynomial may have irrational or complex roots. The theorem only identifies possible rational roots.

Q3: How do I know which of these possible roots are actual roots?
A: You need to test each possible root by substituting into the polynomial or using synthetic division.

Q4: Does this work for polynomials of any degree?
A: Yes, the theorem applies to polynomials of degree 1 or higher, though it's most useful for degree 3 and higher.

Q5: What about repeated roots?
A: The calculator shows each possible root only once, even if it might be a repeated root of the polynomial.