1. What is the Rational Zeros Theorem?

The Rational Zeros Theorem provides a complete list of possible rational zeros (roots) of a polynomial function with integer coefficients. It states that if a polynomial has a rational zero, it must be 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 Zeros Theorem formula:

Where:

• \( p \) — Constant term of the polynomial
• \( q \) — Leading coefficient of the polynomial
• Factors include both positive and negative values

Explanation: The theorem helps narrow down the possible rational solutions to a polynomial equation, which can then be tested using synthetic division or other methods.

3. Importance of Rational Zeros

Details: Finding rational zeros is a crucial step in solving polynomial equations, factoring polynomials, and graphing polynomial functions. It provides a systematic way to identify potential solutions.

4. Using the Calculator

Tips: Enter the constant term (p) and leading coefficient (q) of your polynomial. The calculator will display all possible rational zeros based on the factors of these numbers.

5. Frequently Asked Questions (FAQ)

Q1: Does the theorem guarantee a rational zero exists?
A: No, it only lists possible rational zeros if any exist. The polynomial might have only irrational or complex zeros.

Q2: What if the polynomial has non-integer coefficients?
A: The theorem only applies to polynomials with integer coefficients. For other cases, different methods must be used.

Q3: How do I know which possible zero is actually a zero?
A: You need to test each possible zero by substitution or using synthetic division to verify if it's indeed a root.

Q4: What about repeated factors?
A: The calculator automatically removes duplicate values in the final list of possible zeros.

Q5: Can this be used for higher-degree polynomials?
A: Yes, the theorem applies to polynomials of any degree as long as they have integer coefficients.