If d d x [(x+1) (x^2+1 ) (x^4+1 ) (x^8+1 ) ]= (15 x^p-16 x^q+1 )(x-1)⁻² , then (p, q) is equal to

Step-by-step solution

aligned & d d x [(x+1) (x^2+1 ) (x^4+1 ) (x^8+1 ) ] &= (15 x^ -16 x^9+1 ) (x-1)^2 & LHS = d d x [ (x^2-1 ) (x^2+1 ) (x^4+1 ) (x^8+1 ) (x-1) ] &= d d x [ (x^4-1 ) (x^4+1 ) (x^8+1 ) (x-1) ] &= d d x [ (x^8-1 ) (x^8+1 ) (x-1) ] &= d d x [ (x¹⁶-1 ) (x-1) ] &= (x-1) (16 x¹⁵ )- (x¹⁶-1 ) (x-1)^2 &= 16 x¹⁶-16 x¹⁵-x¹⁶+1 (x-1)^2 aligned = 15 x¹⁶-16 x¹⁵+1 (x-1)^2 On comparing LHS = RHS, we get p=16 and q=15 (p, q)=(16,15)