A Cubic Polynomial Transformed

Algebra Level 4

The Roots of the following Cubic Equation are \(\displaystyle \alpha\), \(\displaystyle \beta\) and \(\displaystyle \gamma\).

\[\displaystyle x^3+7x^2+2x+9=0\]

Then the value of the Expression

\[\displaystyle \dfrac{(3 \alpha-2)(3 \beta-2)(3 \gamma-2)}{(2+3 \alpha)(2+3 \beta)(2+3 \gamma)}\]

Can be written as \(\displaystyle \dfrac{a}{b}\), where \(\displaystyle a\) and \(\displaystyle b\) are coprime positive integers. Find \(\displaystyle a+b\).

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