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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Expert Answer

Arham Warner
Answered 2021-09-15 Author has 24174 answers
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, \(\displaystyle{g{{\left({x}\right)}}}+{h}{\left({x}\right)}={2}{x}^{{2}}+{4}{x}+{1}\)
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