Factorise: $1 - x^{3}$

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Solution

Using the identity $a^{3} - b^{3} = (a - b) (a^{2} + b^{2} + a b)$

Using the above identity, the expression $1 - x^{3}$ can be factorised as follows:

$1 - x^{3} = (1)^{3} - (x)^{3} = (1 - x) [(1)^{2} + x^{2} + (1 \times x)] = (1 - x) (x^{2} + 1 + x)$

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

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$1) x^{3} - 2 x^{2} - x + 2$
$2) 2 y^{3} + y^{2} - 2 y - 1$

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