Question

How do you factor `x^8 - 1`?

Answer 1

`(x^4 - 1)(x^4 + 1)`

Explanation:

Using the difference of squares equation, `a^2 - b^2 = (a - b) (a + b)`
I can determine the answer based on the facts that `x^8` can be formatted as a square, `(x^4)^2`, and that 1 is a square (for itself). Because of these, I am able to use the difference of squares equation. I take the `a` value, `x^4`, and take the `b` value, 1, and plug them into the equation.

If we wanted to check this, we would need to expand the answer that we have gained using FOIL (First Outer Inner Last).
I will demonstrate the steps below:
`(x^4 - 1)(x^4 + 1)`
First:
`x^8 = (x^4)(x^4)`
Outer:
`x^4 = (x^4)(1)`
Inner:
`-x^4 = (x^4)(-1)`
Last:
`-1 = (-1)(1)`
Final expanded form:
`x^8 + x^4 - x^4 - 1`
Which can then cancel the `x^4` to get our original equation, confirming the answer is correct.