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 :
Step-by-step process:
- Simplify
The expression contains a division of by . Perform polynomial long division for :
- Divide by , which gives: This implies: Thus, our expression becomes:
- Compute modulo The polynomial is irreducible over and can serve as a modulus. Since our expression involves a fractional term , the computation must occur in the field .
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:
- Do you want the solution expressed in a finite field such as ?
- Should we perform numerical substitution at specific values of ?
- Do you want me to verify if the given modulus polynomial is irreducible?
- Would you like the computation visualized step-by-step with a detailed polynomial division?
- 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
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