Factorise: $y^{4} - 2 y^{3} + y - 2$

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Solution

The equation $y^{4} - 2 y^{3} + y - 2$ can be factorised as follows:

$y^{4} - 2 y^{3} + y - 2 = (y^{4} - 2 y^{3}) + (y - 2)$
$= y^{3} (y - 2) + 1 (y - 2)$
$= (y^{3} + 1) (y - 2)$
$= (y + 1) (y^{2} - y + 1) (y - 2)$ { $∵ a^{3} + b^{3} = (a + b) (a^{2} + b^{2} - a b$ }

Hence, $y^{4} - 2 y^{3} + y - 2 = (y + 1) (y^{2} - y + 1) (y - 2)$

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