Question

Factor the expression completely over the complex numbers.

x^4 − 625


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Answer 1

(x-5)(x+5)(x-5i)(x+5i)

Explanation:

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Answer 2

You have to observe that
`x^4` and 625 are both squares (of
`x^2` and 25, respectively). So, you can use the "difference of square" pattern for factorization:

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

to write

`x^4 - 625 = (x^2+25)(x^2-25)`

Note that, again,
`x^2-25` is a difference of square:

`x^2-25 = (x+5)(x-5)`

On the other hands,
`x^2+25` admits no factorization, because it's a second-degree polynomial (thus a parabola) with no solutions.

So, the whole expression becomes

`x^4 - 625 = (x^2+25)(x+5)(x-5)`