How do you factor $z^3 - 4$ ?

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Answer 1 · 2016-04-26T21:35:55

$z^3-4 = (z-root(3)(4))(z^2+root(3)(4)z+2root(3)(2))$

This can be factored as a difference of cubes.

The difference of cubes identity can be written:

$a^3-b^3 = (a-b)(a^2+ab+b^2)$

We can use this with $a=z$ and $b=root(3)(4)$ as follows:

$z^3-4$

$= z^3-(root(3)(4))^3$

$=(z-root(3)(4))(z^2+root(3)(4)z+root(3)(16))$

$=(z-root(3)(4))(z^2+root(3)(4)z+2root(3)(2))$

How do you factor z^3 - 4z^3 - 4?