List All Rational Zeros Calculator

Rational Zero Theorem:

\[ \text{If } P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_0 \text{ has integer coefficients, then every rational zero of } P \text{ is of the form } \pm \frac{p}{q} \text{ where:} \] \[ p \text{ divides the constant term } a_0 \] \[ q \text{ divides the leading coefficient } a_n \]

Possible Rational Zeros:

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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 any possible rational zero of a polynomial is a fraction ±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:

\[ \text{For } P(x) = a_nx^n + \cdots + a_0, \text{ possible zeros are } \pm \frac{p}{q} \text{ where } p \mid a_0 \text{ and } q \mid a_n \]

Steps:

Identifies the constant term (a₀) and leading coefficient (aₙ)
Finds all factors of both numbers
Generates all possible ±p/q combinations
Simplifies fractions and removes duplicates

3. Importance of Finding Rational Zeros

Details: Finding rational zeros helps in factoring polynomials, solving polynomial equations, and graphing polynomial functions. It's a crucial first step in polynomial analysis.

4. Using the Calculator

Tips: Enter the polynomial coefficients as comma-separated integers from highest degree to lowest. Example: "2,-3,-11,6" for 2x³ - 3x² - 11x + 6.

5. Frequently Asked Questions (FAQ)

Q1: Does this list guarantee actual zeros?
A: No, it only lists possibilities. You must test each candidate to verify if it's actually a zero.

Q2: What if my polynomial has non-integer coefficients?
A: Multiply through by the LCD to convert to integer coefficients before using the theorem.

Q3: How do I test if a possible zero is actual?
A: Use synthetic division or substitution to check if P(p/q) = 0.

Q4: What about irrational or complex zeros?
A: This theorem only identifies rational zeros. Other methods are needed for other types.

Q5: Can there be no rational zeros?
A: Yes, many polynomials have only irrational or complex zeros.