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Higher Order Equations

The number of...

Question

The number of common roots of the two equations $x^{3} - x^{2} + x - 1 = 0$ and $x^{4} - 1 = 0$ is

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Solution

$x^{3} - x^{2} + x - 1 = 0 \Rightarrow (x - 1) (x^{2} + 1) = 0 \Rightarrow x = 1, i, - i$
$x^{4} - 1 = 0 \Rightarrow (x + 1) (x - 1) (x^{2} + 1) = 0 \Rightarrow x = - 1, 1, i, - i$
$∴$ Number of common roots $= 3$

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