"Sum" of the functions are interesting

Algebra Level 3
  • \(f_k (x)\) has three roots and they are \(k\), \(-k\) and \(0\), where \(k\) is a real positive integer \(\geq 1\). Which of the followings (in terms of \(n\)) is the value of \(x\) when\( f_1 (x)+f_2 (x)+\cdots+f_n (x)=0\)?
  • Assume all \(f_k (x)\) has equal real constant \(d\), such that \(f_k (x)=d(x)(x-k)(x+k)\) where \(d\) is not \(0\)
  • Given that \(x>0\) and \(n=24\),
  • express your answer in the form \( \dfrac{a \sqrt b}c \), where \(a,b\) and \(c\) are positive integers with \(a,c\) coprime and \(b\) square-free.

Submit your answer in the form \(a+b+c\).


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