First slide

Theory of equations

Question

If $α, β$ the roots of $x^{2} + px + q = 0 and x^{2 n} + p^{n} x^{n} + q^{n} = 0$ and if $(α / β), (β / α)$ are the roots of $x^{n} + 1 + (x + 1)^{n} = 0$ then $n (\in N)$

Moderate

A

must be an odd integer
B

may be any integer
C

must be an even integer
D

cannot say anything

Solution

We have
$α + β = - p and αβ = q (1)$
Also, since $α, β$ are the roots of $x^{2 n} + p^{n} x^{n} + q^{n} = 0$ ,
We have $α^{2 n} + p^{n} α^{n} + q^{n} = 0 and β^{2 n} + p^{n} β^{n} + q^{n} = 0$
Subtracting the above relations, we get
$(α^{2 n} - β^{2 n}) + p^{n} (α^{n} - β^{n}) = 0$
$∴ α^{n} + β^{n} = - p^{n} (2)$
Given, $α / β or β / α is a root of x^{n} + 1 + (x + 1)^{n} = 0$ . So,
$\begin{array}{l} (α / β)^{n} + 1 + [(α / β) + 1]^{n} = 0 \\ \Rightarrow (α^{n} + β^{n}) + (α + β)^{n} = 0 \\ \Rightarrow - p^{n} + (- p)^{n} = 0 [Using (1) and (2)] \end{array}$
It is possible only when n is even.

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