Rational Zeros Calculator

Rational Zeros Theorem:

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

Possible ± Factors p:

Possible ± Factors q:

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:

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

Steps:

Identifies all factors of the constant term (p)
Identifies all factors of the leading coefficient (q)
Forms all possible combinations of p/q
Simplifies 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 from highest degree to lowest, separated by commas. For example, for 2x³ - 3x² - 11x + 6, enter "2,-3,-11,6".

5. Frequently Asked Questions (FAQ)

Q1: Does the calculator guarantee all zeros are rational?
A: No, it only lists possible rational zeros. The polynomial might have irrational or complex zeros.

Q2: What if my polynomial has non-integer coefficients?
A: Multiply by the least common denominator to convert to integer coefficients first.

Q3: How do I test which zeros are actual zeros?
A: Use synthetic division or substitution to verify each possible zero.

Q4: Why are negative factors included?
A: The theorem considers both positive and negative factors since (-p)/q = -(p/q).

Q5: What if the leading coefficient is 1?
A: Then q can only be ±1, so possible zeros are simply the factors of the constant term.