Let f(x) be a cubic polynomial such that f(0)=0 , f(1)=3 , f(2)=14 , and f(3)=39 . The value of _ x=1 ¹⁰ f(x) is

Step-by-step solution

Let f(x) = ax^3 + bx^2 + cx + d . Since f(0) = 0 , we have d = 0 . Using the given values: f(1) = a + b + c = 3 f(2) = 8a + 4b + 2c = 14 4a + 2b + c = 7 f(3) = 27a + 9b + 3c = 39 9a + 3b + c = 13 Subtracting the first equation from the second: 3a + b = 4 Subtracting the second equation from the third: 5a + b = 6 Solving these two equations gives 2a = 2 a = 1 . Then b = 4 - 3(1) = 1 . And c = 3 - 1 - 1 = 1 . Thus, f(x) = x^3 + x^2 + x . Now, we need to find _ x=1 ¹⁰ f(x) : _ x=1 ¹⁰ (x^3 + x^2 + x) = _ x=1 ¹⁰ x^3 + _