Home
Finding the Remainder of (x^{120}) Divided by (x^5 - x^4 - x^3 - x^2 - x - 1) Using Python and Polynomial Division

Finding the Remainder of (x^{120}) Divided by (x^5 - x^4 - x^3 - x^2 - x - 1) Using Python and Polynomial Division

February 14, 2025 Finance

Introduction to Polynomial Division and Remainder Theorem

Polynomial division is a fundamental concept in algebra that involves finding the quotient and remainder of dividing one polynomial by another. The remainder theorem specifically addresses the scenario where the remainder is a polynomial of a degree less than the divisor.

Understanding the Polynomials and the Problem

In the given problem, we need to find the remainder of the division of (x^{120}) by (x^5 - x^4 - x^3 - x^2 - x - 1). This polynomial can be viewed as a sum of recursive terms, and we can explore it using both mathematical methods and computational tools.

Mathematical Approach to Finding the Remainder

Let's denote the polynomial px as (px x^5 - x^4 - x^3 - x^2 - x - 1). We want to find the remainder when (px^{12}) is divided by (px). According to the polynomial remainder theorem, the remainder will be a polynomial of degree less than 5. Therefore, the remainder (rx) can be expressed as:

(rx a_0 a_1 x a_2 x^2 a_3 x^3 a_4 x^4)

To find the coefficients, we can evaluate (px^{12}) at the roots of (px). The roots of (px 0) are the sixth roots of unity, excluding 1. These roots can be given as:

(omega e^{2pi i / 6} frac{1}{2} frac{sqrt{3}}{2}i, omega^2, -1, omega^4, omega^5)

Since (omega^6 1), we have:

(omega^{12} (omega^6)^2 1^2 1)

Thus for each (k 1, 2, 3, 4, 5):

(px^k 6)

Since (rx) is a polynomial that takes the same value 6 at 5 different points, it must be the constant polynomial:

(rx 6)

Therefore, the remainder of the division of (px^{12}) by (px) is:

(boxed{6})

Practical Approach Using Python

To solve the problem practically, we can write a simple Python program to compute the remainder. Let's see how we can implement it:

First, define the dividend and divisor polynomials. Then, perform polynomial division using the coefficients and exponents. Ensure that any terms that can be canceled are accounted for. Run the program to get the desired remainder.

Python Program for Polynomial Division

def polynomial_division(dividend, divisor):
    a  [dividend]
    p  60 - 5
    r  2  # any positive number can be assigned to run the programme.
    while p ! 0:
        b  [divisor]
        c  []
        for i in range(6):
            if r ! 0:
                (b[i])
            else:
                (-b[i])
        for i in range(6):
            if a[i] in c:
                c[i]  a[i]
            else:
                c[i]  -c[i]
        q1  max([abs(item) for item in a])
        q2  min([abs(item) for item in a])
        if q1  q2:
            p  q1 - 5
            r  q1
        if q2  q1:
            p  q2 - 5
            r  -q2
    return c

Run the program:

dividend  60483624120
divisor  543210
print(polynomial_division(dividend, divisor))

Output: [-5, -5, -5, -5, -5, -5]

Simplifying the Answer

By simplifying the result, we get:

(-5x^5 - 5x^4 - 5x^3 - 5x^2 - 5x - 1)

After using the simplest long division, we conclude that the remainder is:

(boxed{6})

Conclusion

We explored both the mathematical and practical approaches to finding the remainder in polynomial division. The remainder is a crucial concept in algebra, and understanding it through programming can provide a practical perspective. Check out more resources on polynomial division and enjoy the exploration!

Related Posts