high school precalc math question?

Question

find the cubic polynomial p(x) such that p(1) + p(2) + ... + p(n) = n^4

Answer 1

You're looking to find a, b, c, and d such that

a * sum (k^3) + b * sum (k^2) + c * sum (k) + d * sum (1) = n^4, where in each case the sum goes from 1 to n.

sum(1) = n
sum (k) = n(n+1)/2
sum (k^2) = some cubic expression in n that I forget
sum (k^3) = some quartic expression in n that I forget

I'm guessing you know what the expressions are that I've forgotten.

In essence you have four linear equations in the four variables a, b, c, and d, by setting the coefficient of n^4 equal to 1 and the coefficients of n^3, n^2, and n equal to 0.

What's more, it's a VERY easy system to solve, because only one equation mentions a, only two mention b, and so on.

So that's how to do the problem. :)