If _ x 1 x^4-1 x-1 = _ x k x^3-k^3 x^2-k^2 , where k 0 , then what is the value of k ?

Step-by-step solution

aligned _ x 1 x^4-1 x-1 & = _ x 1 (x-1)(x+1) (x^2+1 ) (x-1) & = _ x 1 (x+1) (x^2+1 )=(1+1)(1+1)=4 aligned Now, _ x k x^3-k^3 x^2-k^2 =4 _ x k (x-k) (x^2+k^2+x k ) (x-k)(x+k) =4 aligned & _ x k (x^2+k^2+x k ) (x+k) =4 & k^2+k^2+k k k+k =4 3 2 k=4 k= 8 3 aligned