If $tan A, tan B, tan C$ are the roots of the equation $2 x^{3} - 5 x^{2} - 2 x + 4 = 0,$ then the value of $tan (A + B + C)$ is

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Solution

Given, $tan A, tan B, tan C$ are the roots of the equation $2 x^{3} - 5 x^{2} - 2 x + 4 = 0$
So,
$\sum tan A = tan A + tan B + tan C = \frac{5}{2}$
$\sum tan A tan B = tan A tan B + tan B tan C + tan C tan A = - 1$
$tan A tan B tan C = - 2$

Now,
$tan (A + B + C) = \frac{\sum tan A - tan A tan B tan C}{1 - \sum tan A tan B} = \frac{\frac{5}{2} - (- 2)}{1 + 1} = \frac{9}{4}$

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