Sum of polynomials over a ring

Question

Please refer to question 1. in the below link (pg 469)

Ques link

I think the question is incorrect as $f(x)$ contains the term $-5$ which doesn't belong to the ring $Z8[x]$ or I should go ahead by changing $f(x)$ to $f(x)= 4x+3$ and similarl changes for $g(x)$ and them add both using modulus funtion ?

Answer 1

$-5$ is an element of $\mathbb{Z}_8$. As you mentioned we have $-5=3=11=19=27=-13=-21=...$ in $\mathbb{Z}_8$.

$f(x)=4x-5=4x+3$

$g(x)=2x^2-4x+2=2x^2+4x+2=2(x^2+2x+1)=2(x+1)^2$

We get:

$(f+g)(x)=4x+3+2x^2+4x+2=2x^2+8x+5=2x^2+5$

and

$(f\cdot g)(x)=(4x+3)(2(x+1)^2)=8x(x+1)^2+6(x+1)^2\stackrel{8=0}{=}6(x+1)^2$