ABC's are hauntin' you

Algebra Level 4

If \[\Large \color{red}{a}+\color{blue}{b}+\color{green}{c} = 0,\] then \[\ \Large \dfrac{\color{red}{a}^5 + \color{blue}{b}^5 + \color{green}{c} ^5}{\color{red}{a}\color{blue}{b} + \color{blue}{b}\color{green}{c} +\color{green}{c} \color{red}{a}}=K\cdot \color{red}{a}\color{blue}{b}\color{green}{c}.\] \[\Large K \ = \ ?\]

Note: Assume that \(a,b,c\) are not all 0.

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