First slide

Theory of equations

Question

If $α, β and γ$ are the roots of $x^{3} + 8 = 0$ , then the equation whose roots are $α^{2}, β^{2} and γ^{2}$ is

Moderate

A

$x^{3} - 8 = 0$
B

$x^{3} - 16 = 0$
C

$x^{3} + 64 = 0$
D

$x^{3} - 64 = 0$

Solution

Let $y = x^{2} . Then x = \sqrt{y}$
$∴ x^{3} + 8 = 0 \Rightarrow y^{3 / 2} + 8 = 0$
$\Rightarrow y^{3} = 64 \Rightarrow y^{3} - 64 = 0$
Thus the equation having roots $α^{2}, β^{2} and γ^{2}$ is $x^{3} - 64 = 0$

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