Question

How do you find the complex roots of `t^4-1=0`?

Answer 1

`t^4-1=0` has roots `+-1` and `+-i`

In a sense these are all complex roots, but `+-i` particularly so, in that they are not Real roots.

Explanation:

The difference of squares identity can be written:

`a^2-b^2 = (a-b)(a+b)`

We find:

`0 = t^4-1`

`color(white)(0) = (t^2)^2 - 1^2`

`color(white)(0) = (t^2 - 1)(t^2 + 1)`

`color(white)(0) = (t^2 - 1^2)(t^2 - i^2)`

`color(white)(0) = (t - 1)(t+1)(t- i)(t+i)`

Hence `t = +-1` or `t = +-i`

The roots `+-i` are non-Real Complex roots.

The roots `+-1` are Real roots. Since Real numbers are a subset of Complex numbers, `+-1` are also Complex roots.