(1) If (x⁸-1)f(x)+(x⁵-1)g(x) = x-1,

Then find the value of f(x) and g(x). 

well..

it can be written as,for x !=1

(1+x+x^2+x^3...x^7)f(x) +(1+x+x^2+x^3 +x^4 )g(x) =1

so (1+x+x^2+x^3 +x^4 )(g(x)+f(x)) + f(x)(x^5+x^6+x^7) =1

so it shud be such tht the f(x) and g(x) are ble to remove the x terms from the LHS..

so i wud say tht it wud be lik f(x)=1/(x^5+x^6+x^7) and g(x)= -f(x) but ofcourse this aint rigorous proof..more of intution..