Find pair of polynomials a(x) and b(x) If $a(x) + b(x) = x^6-1$ and $\gcd(a(x),b(x))=x+1$ then find a pair of polynomials of $a(x)$,$b(x)$.
Prove or disprove, if there exists more than 1  more distinct values of the polynomials.

Since $\gcd\{a(x),b(x)\}=x+1$, there are polynomials $p(x)$ and $q(x)$ such that $a(x)=(x+1)p(x)$ and $b(x)=(x+1)q(x)$. 
$$x^6-1=(x+1)\left(x^5-x^4+x^3-x^2+x-1\right)\;,$$
so 
$$p(x)+q(x)=x^5-x^4+x^3-x^2+x-1\;.\tag{1}$$
You can easily find several pairs of relatively prime polynomials $p(x)$ and $q(x)$ that satisfy $(1)$, and they give you pairs $a(x),b(x)$.
