Let P(x) be a polynomial of degree 3 such that it satisfies the identity P(x) + P'(x) + P''(x) = x^3 + 5x^2 + 13x + 11 for all real x . The value of P(2) is equal to:

Step-by-step solution

Let the cubic polynomial be P(x) = ax^3 + bx^2 + cx + d . The derivatives are: P'(x) = 3ax^2 + 2bx + c P''(x) = 6ax + 2b Adding these gives: P(x) + P'(x) + P''(x) = ax^3 + (b + 3a)x^2 + (c + 2b + 6a)x + (d + c + 2b) We are given that this sum equals x^3 + 5x^2 + 13x + 11 . Equating the corresponding coefficients: Coefficient of x^3 : a = 1 Coefficient of x^2 : b + 3a = 5 b + 3(1) = 5 b = 2 Coefficient of x : c + 2b + 6a = 13 c + 2(2) + 6(1) = 13 c + 10 = 13 c = 3 Constant term: d + c + 2b = 11 d + 3 + 2(2) = 11 d +