First slide

Theory of expressions

Question

If $α, β, γ$ are the roots of $x^{3} + 2 x^{2} - 3 x - 1 = 0$ , then $α^{- 2} + β^{- 2} + γ^{- 2}$ is equal to

Moderate

A

12
B

13
C

14
D

15

Solution

Here, $α + β + γ = - 2$ …………....(i)
$αβ + βγ + γα = - 3$ .…………….(ii)
and $αβγ = 1$ …………………....(iii)
On squaring Eq. (ii), we get
$\begin{aligned} α^{2} β^{2} + β^{2} γ^{2} + γ^{2} α^{2} + 2 αβγ (α + β + γ) = 9 \\ \Rightarrow α^{2} β^{2} + β^{2} γ^{2} + γ^{2} α^{2} = 9 - 2 (1) (- 2) = 13 \end{aligned}$
Now, $α^{- 2} + β^{- 2} + r^{- 2} = \frac{β^{2} γ^{2} + γ^{2} α^{2} + α^{2} β^{2}}{(αβγ)^{2}} = \frac{13}{1} = 13$

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