How do you factor x^6 - 1?

1 Answer
Jul 2, 2015

x^6-1 = (x^3-1)(x^3+1)

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

Explanation:

Use difference of squares, difference of cubes and sum of cubes:

[1]: a^2-b^2 = (a-b)(a+b)

[2]: a^3-b^3 = (a-b)(a^2+ab+b^2)

[3]: a^3+b^3 = (a+b)(a^2-ab+b^2)

x^6-1

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

=(x^3-1)(x^3+1) by [1]

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

=(x-1)(x^2+x+1)(x+1)(x^2-x+1) by [2] and [3]