How do you factor $r^3 - 1$ ?

Question

7185 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-12-20T00:32:58

Use the difference of cubes identity to find:

$r^3-1 = (r-1)(r^2+r+1)$

The difference of cubes identity can be written:

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

Use this with $a=r$ and $b=1$ as follows:

$r^3-1$

$=r^3-1^3$

$=(r-1)(r^2+(r)(1)+1^2)$

$=(r-1)(r^2+r+1)$

If you allow Complex coefficients then this can be factored a little further:

$=(r-1)(r-omega)(r-omega^2)$

where $omega = -1/2+sqrt(3)/2 i$ is the primitive Complex cube root of $1$ .

How do you factor r^3 - 1r^3 - 1?