For the equation, $x^{7} + 3 x^{6} - 2 x^{5} + 4 x^{4} - 4 x^{3} + 3 x^{2} - 2 x + 1 = 0,$
the value of the sum of the roots taken $3$ at a time is:

- 4

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Solution

The correct option is A $- 4$
The given equation is $x^{7} + 3 x^{6} - 2 x^{5} + 4 x^{4} - 4 x^{3} + 3 x^{2} - 2 x + 1 = 0.$
According to the relation between the roots and coefficients of an equation $a_{0} x^{n} + a_{1} x^{n - 1} + a_{2} x^{n - 2} + . . . + a_{n} = 0$ with degree $n,$ the sum of the roots taken $r$ at a time is given by
$S_{r} = (- 1)^{r} \frac{a_{r}}{a_{0}}$
$\Rightarrow S_{3} = - \frac{4}{1} = - 4$

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