The Integer Root Theorem is vital for finding roots of polynomial equations with integer coefficients. It states that any root of a polynomial P(x) = a_n x^n + a_(n-1) x^(n-1) + … + a_1 x + a_0 must be a factor of the constant term a_0, and the leading coefficient a_n must divide any potential root. This calculator uses the theorem to predict potential roots based on the polynomial’s coefficients, providing a list of all possible integer roots.
Formula of Integer Root Theorem Calculator
To use the Integer Root Theorem effectively, follow these detailed steps:
- Identify the Polynomial: Start with the polynomial equation P(x) = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0.
- Factors of the Constant Term: Calculate all factors of the constant term a_0.
- Factors of the Leading Coefficient: Similarly, determine the factors of the leading coefficient a_n.
- Calculate Possible Roots: The possible integer roots are the ratios of the factors of a_0 to the factors of a_n.
User-Friendly Table
Here’s a table designed to help users apply the Integer Root Theorem without performing detailed calculations each time:
| Polynomial | Potential Roots |
|---|---|
| x^2 - 5x + 6 | 1, 2, 3 |
| x^3 - 3x^2 + 2x | 0, 1, 2 |
This table serves as a quick reference to understand typical results from common polynomials.
Example of Integer Root Theorem Calculator
Consider the polynomial x^2 - 7x + 10. Applying the Integer Root Theorem:
- Factors of 10 (constant term): ±1, ±2, ±5, ±10
- Factors of 1 (leading coefficient): ±1
- Possible roots based on the theorem: ±1, ±2, ±5, ±10
- By substitution into the polynomial, we find that 2 and 5 are actual roots.
Most Common FAQs
The calculator provides all potential roots based on mathematical laws, ensuring high accuracy in predicting valid roots.
Yes, the Integer Root Theorem Calculator is capable of handling polynomials of any degree as long as the coefficients are integers.
No, users can input as many coefficients as needed, from simple linear equations to complex higher-degree polynomials.