Let P(x) = a₀ + a₁ x + a₂ x^2 + + a₁₃ x¹³ be a polynomial such that x^2 P(x) = (1+x)¹⁵ - 1 - 15x . The value of a₁ + a₃ + a₅ + + a₁₃ is equal to:

Step-by-step solution

Given the polynomial identity: x^2 P(x) = (1+x)¹⁵ - 1 - 15x Using the binomial expansion for (1+x)¹⁵ , we have: (1+x)¹⁵ = ¹⁵C₀ + ¹⁵C₁ x + _ k=2 ¹⁵ ¹⁵C_ k x^k Substitute this into the identity: x^2 P(x) = _ k=2 ¹⁵ ¹⁵C_ k x^k Dividing by x^2 for x 0 , we get: P(x) = _ k=2 ¹⁵ ¹⁵C_ k x^ k-2 Let j = k-2 . Then the polynomial can be written as: P(x) = _ j=0 ¹³ ¹⁵C_ j+2 x^j Comparing this with P(x) = a₀ + a₁ x + a₂ x^2 + + a₁₃ x¹³ , we find that the coefficient a_j is given by: a_j = ¹⁵C_ j+2 We need to find the sum of th