First slide

Theory of equations

Question

Let $α$ and $β$ be the roots of the equation $x^{2} + x + 1 = 0$ . The equation whose roots are $α^{19}, β^{7}$ is

Moderate

A

$x^{2} - x - 1 = 0$
B

$x^{2} - x + 1 = 0$
C

$x^{2} + x - 1 = 0$
D

$x^{2} + x + 1 = 0$

Solution

Given $x^{2} + x + 1 = 0$
$∴ x = \frac{1}{2} [- 1 \pm i \sqrt{3}] = \frac{1}{2} (- 1 + i \sqrt{3}), \frac{1}{2} (- 1 - i \sqrt{3}) = ω, ω^{2} .$
But $α^{19} = ω^{19} = ω and β^{7} = ω^{14} = ω^{2}$ .
Hence the equation will be same.

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