Find Zeros

Rational Zero Theorem:

\[ \text{If } P(x) = a_nx^n + \cdots + a_0 \text{ has integer coefficients, then every rational zero of P is of the form } \frac{p}{q} \text{ where p divides } a_0 \text{ and q divides } a_n \]

Possible Rational Zeros:

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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 if a polynomial has a rational zero p/q, then p must divide the constant term and q must divide the leading coefficient.

2. How Does the Calculator Work?

The calculator uses the Rational Zero Theorem:

\[ \text{If } P(x) = a_nx^n + \cdots + a_0 \text{ has integer coefficients, then every rational zero of P is of the form } \frac{p}{q} \text{ where p divides } a_0 \text{ and q divides } a_n \]

Where:

\( a_0 \) — Constant term of the polynomial
\( a_n \) — Leading coefficient of the polynomial
\( p \) — Factor of the constant term \( a_0 \)
\( q \) — Factor of the leading coefficient \( a_n \)

Explanation: The calculator finds all factors of the constant term and leading coefficient, then generates all possible combinations of p/q (both positive and negative) in reduced form.

3. Importance of Finding Rational Zeros

Details: Finding rational zeros is the first step in solving polynomial equations. These zeros can be tested using synthetic division to factor the polynomial completely.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: Does this find all zeros of a polynomial?
A: No, it only finds possible rational zeros. There may be irrational or complex zeros that this theorem doesn't identify.

Q2: What if my polynomial has non-integer coefficients?
A: The Rational Zero Theorem only applies to polynomials with integer coefficients. You may need to multiply through by a common denominator first.

Q3: How do I know which of these possible zeros are actual zeros?
A: You need to test each possible zero in the polynomial (using substitution or synthetic division) to see which ones yield P(x) = 0.

Q4: Why do we include both positive and negative possibilities?
A: Because a polynomial can have zeros on either side of the x-axis. For example, (x-2)(x+3) has zeros at 2 and -3.

Q5: What if my constant term or leading coefficient is zero?
A: The theorem doesn't apply in these cases. You should factor out x first until you have a non-zero constant term.