Rewrite 1 as `1^4` (one raised to the power of four),
`x^4 - 1^4`
You can write `x^4` as `(x^2)^2` and `1^4` as `(1^2)^2`,
`(x^2)^2 - (1^2)^2`
Apply the difference of squares formula `a^2 - b^2 = (a + b)(a - b)`,
`(x^2 + 1^2)(x^2 - 1^2)`
Again apply the difference of squares formula on the second parenthesis above,
`(x^2 + 1^2)(x + 1)(x - 1)`
Simplify by rewriting `1^2` as 1,
`(x^2 + 1)(x + 1)(x - 1)`
Thus you have completely factored the expression `x^4 - 1` into `(x^2 + 1)(x + 1)(x - 1)`
`x^4 - 1^4`
You can write `x^4` as `(x^2)^2` and `1^4` as `(1^2)^2`,
`(x^2)^2 - (1^2)^2`
Apply the difference of squares formula `a^2 - b^2 = (a + b)(a - b)`,
`(x^2 + 1^2)(x^2 - 1^2)`
Again apply the difference of squares formula on the second parenthesis above,
`(x^2 + 1^2)(x + 1)(x - 1)`
Simplify by rewriting `1^2` as 1,
`(x^2 + 1)(x + 1)(x - 1)`
Thus you have completely factored the expression `x^4 - 1` into `(x^2 + 1)(x + 1)(x - 1)`
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