A strange equation III

Algebra Level 5

The three real roots of the polynomial \(P(x)=x^{3}+3x^{2}-51x+54\sqrt{3}-107\) can be expressed as:

\[ x_{1}=-a\sqrt{b}\cos(\alpha)-c , x_{2}=a\sqrt{b}\cos(\beta)-c , x_{3}=a\sqrt{b}\cos(\theta)-c,\]

where \(x_{1}<x_{2}<x_{3}\), \(0°<\alpha<90°\), \(0°<\beta<90°\), \(0°<\theta<90°\), such that \(a\), \(b\) and \(c\) are positive integers; and \(\alpha\), \(\beta\) and \(\theta\) are integer degress. Find \(\dfrac{\alpha+\beta+\theta-1}{a+b+c}\).

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