Let the polynomial P(x) be defined as (x + x^2-1 )^7 + (x - x^2-1 )^7 = a₇ x^7 + a₆ x^6 + + a₁ x + a₀ for x > 1 . The value of _ k=1 ^7 k a_k is :

Step-by-step solution

The given polynomial is P(x) = _ k=0 ^7 a_k x^k . Differentiating both sides with respect to x gives: P'(x) = _ k=1 ^7 k a_k x^ k-1 Substituting x = 1 , we get P'(1) = _ k=1 ^7 k a_k . Thus, the required sum is exactly P'(1) . Let us find P'(x) by differentiating the original expression using the chain rule: P'(x) = 7 (x + x^2-1 )^6 (1 + x x^2-1 ) + 7 (x - x^2-1 )^6 (1 - x x^2-1 ) P'(x) = 7 (x + x^2-1 )^6 ( x^2-1 + x x^2-1 ) + 7 (x - x^2-1 )^6 ( x^2-1 - x x^2-1 ) P'(x) = 7 x^2-1 [ (x + x^2-1 )^7 - (x - x^2-1 )^7 ]