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Relations between Roots and Coefficients : Higher Order Equations

If α , β , γ ...

Question

If $α, β, γ$ are the roots of $x^{3} + 3 x^{2} + 4 x + 5 = 0$ , then which of the following is/ are true.

\sum α = 3

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\sum \frac{1}{α} = - \frac{4}{5}

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\sum α^{2} = 1

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\sum β^{2} γ^{2} = - 14

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Solution

The correct option is D $\sum β^{2} γ^{2} = - 14$
$α + β + γ = - 3 α β + β γ + γ α = 4, α β γ = - 5$

$\sum \frac{1}{α} = \frac{α β + β γ + γ α}{α β γ} = - \frac{4}{5}$

$\sum α^{2} = (α + β + γ)^{2} - 2 (α β + β γ + γ α) = 9 - 8 = 1$

$\sum β^{2} γ^{2} = (α β + β γ + γ α)^{2} - 2 α β γ (α + β + γ) = 16 - 2 (- 5) (- 3) = - 14$

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