To add two polynomials [tex]\(f(x) = 1\)[/tex] and [tex]\(g(x) = x^2 + x + 1\)[/tex] over GF(2), we will follow these steps:
1. Understand the Polynomials:
- [tex]\(f(x) = 1\)[/tex]: This means that the polynomial is a constant value of 1.
- [tex]\(g(x) = x^2 + x + 1\)[/tex]: This polynomial contains terms up to [tex]\(x^2\)[/tex] with coefficients in GF(2).
2. Evaluate the Polynomials over GF(2):
- GF(2) means the coefficients are taken modulo 2. The possible values for the inputs [tex]\(x\)[/tex] in GF(2) are [tex]\(0\)[/tex] and [tex]\(1\)[/tex].
3. Addition Rule in GF(2):
- Addition in GF(2) follows the rule: [tex]\(a + b \equiv (a + b) \mod 2\)[/tex], where [tex]\(a, b \in \{0, 1\}\)[/tex].
4. Step-by-Step Addition:
- For [tex]\(x = 0\)[/tex]:
- Evaluate [tex]\(f(x)\)[/tex] at [tex]\(x = 0\)[/tex]: [tex]\(f(0) = 1\)[/tex]
- Evaluate [tex]\(g(x)\)[/tex] at [tex]\(x = 0\)[/tex]: [tex]\(g(0) = 0^2 + 0 + 1 = 1\)[/tex]
- Add the polynomials at [tex]\(x = 0\)[/tex]: [tex]\(f(0) + g(0) = 1 + 1 = 2 \equiv 0 \mod 2\)[/tex]
- For [tex]\(x = 1\)[/tex]:
- Evaluate [tex]\(f(x)\)[/tex] at [tex]\(x = 1\)[/tex]: [tex]\(f(1) = 1\)[/tex]
- Evaluate [tex]\(g(x)\)[/tex] at [tex]\(x = 1\)[/tex]: [tex]\(g(1) = 1^2 + 1 + 1 = 1 + 1 + 1 = 3 \equiv 1 \mod 2\)[/tex] (since [tex]\(3\)[/tex] mod [tex]\(2\)[/tex] is [tex]\(1\)[/tex])
- Add the polynomials at [tex]\(x = 1\)[/tex]: [tex]\(f(1) + g(1) = 1 + 1 = 2 \equiv 0 \mod 2\)[/tex]
5. Compile Results:
- The result of adding [tex]\(f(x) = 1\)[/tex] and [tex]\(g(x) = x^2 + x + 1\)[/tex] over GF(2) at [tex]\(x = 0\)[/tex] and [tex]\(x = 1\)[/tex] is:
- For [tex]\(x = 0\)[/tex], the result is [tex]\(0\)[/tex].
- For [tex]\(x = 1\)[/tex], the result is [tex]\(0\)[/tex].
Hence, the final result of adding these two polynomials over GF(2) is:
[tex]\[ [0, 0] \][/tex]
