How do you factor completely 1+x3?

1 Answer
Dec 20, 2015

Use the sum of cubes identity to find:

1+x3=(1+x)(1x+x2)

Explanation:

The sum of cubes identity may be written:

a3+b3=(a+b)(a2ab+b2)

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

1+x3

=13+x3

=(1+x)(12(1)(x)+x2)

=(1+x)(1x+x2)

The remaining quadratic factor (1x+x2) 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+ωx)(1+ω2x)

or if you prefer:

=(1+x)(ω+x)(ω2+x)

where ω=12+32i is the primitive Complex cube root of 1.