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 \( \mathbb{Z}_{7}[x] \).

Answer

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Answer

\(q(x) = 4x^2 + 4x + 4\), \(r(x) = x + 4\)

Steps

Step 1 ：\(q(x) = x^4 + x^3 + x + 6\), \(r(x) = 2x^2 - x + 1\)

Step 2 ：\(q(x) = 4x^2 + 4x + 4\), \(r(x) = x + 4\)