Problem
1) Find \( q(x) \) and \( r(x) \) as described by the division algorithm. a) \( f(x)=x^{6}+3 x^{5}+4 x^{2}-3 x+2, g(x)=x^{2}+2 x-3 \) in \( \mathbb{Z}_{7}[x] \). b) \( \mathrm{f}(\mathrm{x})=\mathrm{x}^{4}-x^{2}+3 \mathrm{x}+3, g(\mathrm{x})=4 \mathrm{x}^{2}+2 \) in \( \mathbb{Z}_{7}[x] \).
Solution
Step 1 ：\(q(x) = x^4 + x^3 + x + 6\), \(r(x) = 2x^2 - x + 1\)
Step 2 ：\(q(x) = 4x^2 + 4x + 4\), \(r(x) = x + 4\)
