Question

How do you factor completely `1 + x^3`?

Answer 1

Use the sum of cubes identity to find:

`1+x^3 = (1+x)(1-x+x^2)`

Explanation:

The sum of cubes identity may be written:

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

In our example, we have `a=1` and `b=x` as follows:

`1+x^3`

`=1^3+x^3`

`=(1+x)(1^2-(1)(x)+x^2)`

`=(1+x)(1-x+x^2)`

The remaining quadratic factor `(1-x+x^2)` cannot be factored into simpler factors with Real coefficients, but if you want a complete factorisation then you can do it with Complex coefficients:

`=(1+x)(1+omega x)(1+omega^2 x)`

or if you prefer:

`=(1+x)(omega+x)(omega^2 + x)`

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