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Transformation of Roots: Algebraic Transformation

If α, β, γ an...

Question

If α, β, γ and δ are the roots of $x^{4} + 2 x^{3} + 3 x^{2} + 4 x + 5 = 0$ .
Find the equation whose roots are $\frac{1}{α}, \frac{1}{β}, \frac{1}{γ}, \frac{1}{δ}$ .

$x^{4} + 3 x^{3} + 2 x^{2} + 4 x + 5 = 0$

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$x^{4} - 2 x^{3} + 3 x^{2} - 4 x + 5 = 0$

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5 $x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 1 = 0$

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5 $x^{4} - 4 x^{3} + 3 x^{2} - 2 x + 1 = 0$

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Solution

The correct option is C
5 $x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 1 = 0$

Given: $x^{4} + 2 x^{3} + 2 x^{2} + 4 x + 5 = 0 . . . (1)$ have roots as α, β, γ, δ.

Changing $x$ to $\frac{1}{x}$

$\Rightarrow \frac{1}{x^{4}} + \frac{2}{x^{3}} + \frac{3}{x^{2}} + \frac{4}{x} + 5 = 0$

$\Rightarrow 5 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 1 = 0 . . . (2)$

Roots of the equation (2) being the reciprocal of the roots of the equation (1) i.e., $\frac{1}{α}$ , $\frac{1}{β}$ , $\frac{1}{γ}$ , $\frac{1}{δ}$ .
Hence the required equation is
$\Rightarrow 5 x^{4} + 4 x^{3} + 3 x^{2} + 2 x + 1 = 0$

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