Math Problem Statement

can you solve:

(x+1)x^3 + x(x^3 +1) = (mod x^4 +x+1) xx^3 + (x+1)(x^3 +1) = (mod x^4 +x+1) (x+1) + x*(x^3 + x^2) = (mod x^4 +x+1) x + (x+1)*(x^3 + x^2) = (mod x^4 +x+1)

Solution

The expressions you provided seem to involve modular arithmetic with respect to x4+x+1x^4 + x + 1. To clarify, you want to simplify or solve these expressions under the modulus x4+x+1x^4 + x + 1. Let's proceed by simplifying each of these expressions step by step, modulo x4+x+1x^4 + x + 1.

Expression 1:

(x+1)x3+x(x3+1)=mod (x4+x+1)(x+1) \cdot x^3 + x \cdot (x^3 + 1) = \text{mod } (x^4 + x + 1) First, expand both terms on the left-hand side: (x+1)x3=x4+x3(x+1) \cdot x^3 = x^4 + x^3 x(x3+1)=x4+xx \cdot (x^3 + 1) = x^4 + x Now, sum the expanded terms: x4+x3+x4+x=2x4+x3+xx^4 + x^3 + x^4 + x = 2x^4 + x^3 + x Next, apply the modulus x4+x+1x^4 + x + 1:

  • Notice that x4x1mod(x4+x+1)x^4 \equiv -x - 1 \mod (x^4 + x + 1). Thus, substitute x4x^4 with x1-x - 1: 2(x1)+x3+x=2x2+x3+x2(-x - 1) + x^3 + x = -2x - 2 + x^3 + x Simplify: x3x2x^3 - x - 2 So, the first expression simplifies to: x3x2mod(x4+x+1)x^3 - x - 2 \mod (x^4 + x + 1)

Expression 2:

xx3+(x+1)(x3+1)=mod (x4+x+1)x \cdot x^3 + (x+1) \cdot (x^3 + 1) = \text{mod } (x^4 + x + 1) Expand both terms: xx3=x4x \cdot x^3 = x^4 (x+1)(x3+1)=x4+x3+x3+1=x4+2x3+1(x+1) \cdot (x^3 + 1) = x^4 + x^3 + x^3 + 1 = x^4 + 2x^3 + 1 Sum the terms: x4+x4+2x3+1=2x4+2x3+1x^4 + x^4 + 2x^3 + 1 = 2x^4 + 2x^3 + 1 Apply the modulus x4+x+1x^4 + x + 1:

  • Substitute x4x1x^4 \equiv -x - 1: 2(x1)+2x3+1=2x2+2x3+12(-x - 1) + 2x^3 + 1 = -2x - 2 + 2x^3 + 1 Simplify: 2x32x12x^3 - 2x - 1 Thus, the second expression simplifies to: 2x32x1mod(x4+x+1)2x^3 - 2x - 1 \mod (x^4 + x + 1)

Expression 3:

(x+1)+x(x3+x2)=mod (x4+x+1)(x+1) + x \cdot (x^3 + x^2) = \text{mod } (x^4 + x + 1) Expand: x(x3+x2)=x4+x3x \cdot (x^3 + x^2) = x^4 + x^3 Thus, the equation becomes: (x+1)+x4+x3(x + 1) + x^4 + x^3 Apply the modulus x4+x+1x^4 + x + 1: (x+1)+(x1)+x3=x3(x + 1) + (-x - 1) + x^3 = x^3 Thus, the third expression simplifies to: x3mod(x4+x+1)x^3 \mod (x^4 + x + 1)

Expression 4:

x+(x+1)(x3+x2)=mod (x4+x+1)x + (x+1) \cdot (x^3 + x^2) = \text{mod } (x^4 + x + 1) Expand: (x+1)(x3+x2)=x4+x3+x3+x2=x4+2x3+x2(x+1) \cdot (x^3 + x^2) = x^4 + x^3 + x^3 + x^2 = x^4 + 2x^3 + x^2 Thus, the equation becomes: x+x4+2x3+x2x + x^4 + 2x^3 + x^2 Apply the modulus x4+x+1x^4 + x + 1: x+(x1)+2x3+x2=2x3+x21x + (-x - 1) + 2x^3 + x^2 = 2x^3 + x^2 - 1 Thus, the fourth expression simplifies to: 2x3+x21mod(x4+x+1)2x^3 + x^2 - 1 \mod (x^4 + x + 1)

Final Simplifications:

  1. x3x2x^3 - x - 2
  2. 2x32x12x^3 - 2x - 1
  3. x3x^3
  4. 2x3+x212x^3 + x^2 - 1

Would you like further details on how these were derived or how to proceed? Here are some related questions:


  1. How does modular arithmetic apply to polynomials?
  2. What are the practical applications of polynomial modular arithmetic?
  3. Can you explain the process of polynomial division in more detail?
  4. How does polynomial factoring work in modular arithmetic?
  5. What are some common problems where polynomial modular reduction is used?

Tip: In polynomial modular arithmetic, terms of degree higher than the modulus are replaced using the modulus equation, reducing them to lower degree equivalents.

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomial Arithmetic
Modular Arithmetic

Formulas

(a + b)(c + d) = ac + ad + bc + bd
x^4 ≡ -x - 1 mod (x^4 + x + 1)

Theorems

Polynomial Modular Arithmetic
Modulus Reduction Theorem for Polynomials

Suitable Grade Level

Undergraduate Level (Mathematics or Computer Science)