Math Problem Statement

((x^2/x+1) + x) mod (x^6 + x^3 + x^2 +x +1 )

Solution

We aim to simplify the following expression and then compute it modulo the polynomial x6+x3+x2+x+1x^6 + x^3 + x^2 + x + 1:

(x2x+1+x)mod(x6+x3+x2+x+1)\left( \frac{x^2}{x+1} + x \right) \mod \left( x^6 + x^3 + x^2 + x + 1 \right)


Step-by-step process:

  1. Simplify x2x+1+x\frac{x^2}{x+1} + x The expression contains a division of x2x^2 by x+1x+1. Perform polynomial long division for x2x+1\frac{x^2}{x+1}:
    • Divide x2x^2 by x+1x+1, which gives: x2÷(x+1)=x1with remainder 1x^2 \div (x+1) = x - 1 \quad \text{with remainder } 1 This implies: x2x+1=x1+1x+1\frac{x^2}{x+1} = x - 1 + \frac{1}{x+1} Thus, our expression becomes:
    = 2x - 1 + \frac{1}{x+1}$$
  2. Compute modulo x6+x3+x2+x+1x^6 + x^3 + x^2 + x + 1 The polynomial x6+x3+x2+x+1x^6 + x^3 + x^2 + x + 1 is irreducible over F2\mathbb{F}_2 and can serve as a modulus. Since our expression involves a fractional term 1x+1\frac{1}{x+1}, the computation must occur in the field F2(x)\mathbb{F}_2(x).

Let me know if you want me to:

  • Simplify this expression completely using field arithmetic, or
  • Work it out in a numerical example (e.g., over finite fields or reals).

Follow-up Questions:

  1. Do you want the solution expressed in a finite field such as F2\mathbb{F}_2?
  2. Should we perform numerical substitution at specific values of xx?
  3. Do you want me to verify if the given modulus polynomial is irreducible?
  4. Would you like the computation visualized step-by-step with a detailed polynomial division?
  5. Should I evaluate the modular reduction symbolically?

Tip:

In polynomial arithmetic, working with modular reductions requires checking the divisor polynomial's properties (irreducibility, field structure). This can change how the solution is interpreted.

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Math Problem Analysis

Mathematical Concepts

Polynomial Division
Modular Arithmetic
Field Theory

Formulas

x^2 / (x+1)
Modulo operation with polynomials
Irreducible polynomials over finite fields

Theorems

Polynomial division
Irreducibility of the polynomial x^6 + x^3 + x^2 + x + 1 over F2

Suitable Grade Level

Grades 11-12