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

Annette Arroyo

Answered question

2021-09-14

Consider the following polynomials over Z8 where a is written for [a] in Z8:
f(x)=2x3+7x+4,g(x)=4x2+4x+6,h(x)=6x2+3
Find each of the following polynomials with all coefficients in Z8
g(x)+h(x)

Answer & Explanation

Arham Warner

Arham Warner

Skilled2021-09-15Added 102 answers

When adding two polynomials coefficients of like powers are added. With coefficients of polynomials in Z8 again the same approach is taken to add the two polynomials.
While adding it is to be noted that in Z8 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 Z8 so taking it modulo 8 gives 7, that is, 7+8=7 modulo 8. So in Z8,7+8=7.
Given polynomials to be added are g(x)=4x2+4x+6,h(x)=6x2+3. To add them, add the coefficients of like terms and then apply modulo 8.
g(x)+h(x)=(4x2+4x+6)+(6x2+3)
=(4x2+6x2)+4x+(6+3)
=10x2+4x+9
=2x2+4x+1 modulo 8
Hence, g(x)+h(x)=2x2+4x+1

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