How do you factor x^3-27?

1 Answer
Mar 15, 2016

Use the difference of cubes identity to find:

x^3-27 = (x-3)(x^2+3x+9)

Explanation:

Both x^3 and 27=3^3 are perfect cubes. So we can use the difference of cubes identity:

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

with a=x and b=3 as follows:

x^3-27

=x^3-3^3

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

=(x-3)(x^2+3x+9)

This is as far as you can go with Real coefficients. If you allow Complex coefficients then you can factor this a little further:

=(x-3)(x-3omega)(x-3omega^2)

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