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Let's solve the given problems numerically and manually in modular arithmetic. --- ### (i) Add the two polynomials in $\mathbb{Z}_5[x]$: Given: $$ (\overline{3}x^2+\overline{4}x+\overline{1}) + (\overline{3}x^3+\overline{4}x^2+\overline{2}x+\overline{3}) $$ Add corresponding coefficients term by term modulo 5: * Degree 3 term: $$ 0 + \overline{3} = \overline{3} $$ * Degree 2 term: $$ \overline{3} + \overline{4} = \overline{7} = \overline{2} \mod 5 $$ * Degree 1 term: $$ \overline{4} + \overline{2} = _____ _____ _______ ______ _________ ____.
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