Find Rational Zeros Calculator

Rational Root Theorem:

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

Possible Rational Zeros:

Actual Rational Zeros:

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1. What is the Rational Root Theorem?

The Rational Root Theorem provides a complete list of possible rational zeros of a polynomial with integer coefficients. It states that any possible rational zero, expressed in lowest terms p/q, has p as a factor of the constant term and q as a factor of the leading coefficient.

2. How Does the Calculator Work?

The calculator implements the Rational Root Theorem in three steps:

Finds all factors of the constant term (p)
Finds all factors of the leading coefficient (q)
Tests all possible combinations of ±p/q (in reduced form)

Example: For P(x) = 2x³ - 3x² - 11x + 6

Constant term (a₀) = 6 → factors: 1,2,3,6
Leading coefficient (aₙ) = 2 → factors: 1,2
Possible zeros: ±1/1, ±1/2, ±2/1, ±3/1, ±3/2, ±6/1
Actual zeros found: -2, 1/2, 3

3. Importance of Finding Rational Zeros

Details: Finding rational zeros is the first step in factoring polynomials and solving polynomial equations. It's fundamental in algebra, calculus, and many applied mathematics fields.

4. Using the Calculator

Tips: Enter coefficients as comma-separated integers, highest degree first. Example: "1,0,-2" for x² - 2. The calculator will list all possible rational zeros and verify which are actual zeros.

5. Frequently Asked Questions (FAQ)

Q1: What if my polynomial has non-integer coefficients?
A: Multiply through by the LCD to convert to integer coefficients before using the theorem.

Q2: What if no rational zeros are found?
A: The polynomial may have irrational or complex zeros only. Consider numerical methods or graphing.

Q3: How accurate is the zero checking?
A: Uses a tolerance of 1e-10 - values smaller than this are considered zeros.

Q4: Can it handle repeated zeros?
A: Yes, repeated zeros will appear multiple times in the results.

Q5: What's the maximum degree polynomial supported?
A: Technically any degree, but very high degrees may slow down calculation.