 Consider the following polynomials over Z_8 where a is written for [a] in Z_8:f(x)=2x^3+7x+4,g(x)=4x^2+4x+6,h(x)=6x^2+3 Annette Arroyo 2021-09-14 Answered
Consider the following polynomials over $$\displaystyle{Z}_{{8}}$$ where a is written for [a] in $$\displaystyle{Z}_{{8}}$$:
$$\displaystyle{f{{\left({x}\right)}}}={2}{x}^{{3}}+{7}{x}+{4},{g{{\left({x}\right)}}}={4}{x}^{{2}}+{4}{x}+{6},{h}{\left({x}\right)}={6}{x}^{{2}}+{3}$$
Find each of the following polynomials with all coefficients in $$\displaystyle{Z}_{{8}}$$
$$\displaystyle{g{{\left({x}\right)}}}+{h}{\left({x}\right)}$$

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When adding two polynomials coefficients of like powers are added. With coefficients of polynomials in $$\displaystyle{Z}_{{8}}$$ again the same approach is taken to add the two polynomials.
While adding it is to be noted that in $$\displaystyle{Z}_{{8}}$$ the coefficients can take values from 0,1,2,3,4,5,6,7. And after adding the coefficients resulting term is taken modulo 8. For example 7+8=15 but this is not in $$\displaystyle{Z}_{{8}}$$ so taking it modulo 8 gives 7, that is, 7+8=7 modulo 8. So in $$\displaystyle{Z}_{{8}}$$,7+8=7.
Given polynomials to be added are $$\displaystyle{g{{\left({x}\right)}}}={4}{x}^{{2}}+{4}{x}+{6},{h}{\left({x}\right)}={6}{x}^{{2}}+{3}$$. To add them, add the coefficients of like terms and then apply modulo 8.
$$\displaystyle{g{{\left({x}\right)}}}+{h}{\left({x}\right)}={\left({4}{x}^{{2}}+{4}{x}+{6}\right)}+{\left({6}{x}^{{2}}+{3}\right)}$$
$$\displaystyle={\left({4}{x}^{{2}}+{6}{x}^{{2}}\right)}+{4}{x}+{\left({6}+{3}\right)}$$
$$\displaystyle={10}{x}^{{2}}+{4}{x}+{9}$$
$$\displaystyle={2}{x}^{{2}}+{4}{x}+{1}$$ modulo 8
Hence, $$\displaystyle{g{{\left({x}\right)}}}+{h}{\left({x}\right)}={2}{x}^{{2}}+{4}{x}+{1}$$