# Find $f_{1}, f_{2}$ in $\mathbb{Z_{6}}[x]$ such that deg$(f_{1})$ = deg$(f_{2}) = 2$ and deg$(f_{1}+f_{2})=1$

1. Find $f_{1}, f_{2}$ in $\mathbb{Z_{6}}[x]$ such that deg$(f_{1})$ = deg$(f_{2}) = 2$ and deg$(f_{1}+f_{2})=1$
2. Find $g_{1}, g_{2}$ in $\mathbb{Z_{6}}[x]$ such that deg$(g_{1})$ = deg$(g_{2}) = 1$ and deg$(g_{1}.g_{2})=1$
• Your solution to 1. is correct. For 2., look for polynomials of the type $g_1(x):=ax+b$, $g_2(x)=cx+d$, $a\neq\bar0$, $c\neq\bar0$. What must the coefficients satisfy in order for $\mathrm{deg}(g_1\cdot g_2)=1$?
– Ian
Feb 1, 2015 at 0:43
• Thanks. I also did the second one. Feb 1, 2015 at 0:50
• Your solution to 2. is also correct.
– Ian
Feb 1, 2015 at 0:51
• Yay. Thanks again :) Feb 1, 2015 at 0:52
• Perhaps you could post your solution as an answer and accept it. Just include your thought process so that it may be of help to visitors.
– Ian
Feb 1, 2015 at 0:55

(1) Let $f_{1}(x) = \bar{3}x^2+\bar{2}x+\bar{1}$ and $f_{2}(x) = \bar{3}x^2+x+\bar{2}$, which are of degree $2$.
Then, $(f_{1}+f_{2}) = \bar{6}x^2 + \bar{3}x + \bar{3} = \bar{0}x^2 + \bar{3}x + 3$ is of degree $1$.
(2) Let $g_{1}(x) = 3+2x$ and $g_{2}(x) = 2+3x$, which are of degree $1$.
Then, $(g_{1}.g_{2}) = \bar{6} + \bar{13}x = \bar{0} + x$ is of degree $1$.