Factor the following completely

x^4 – 625

call y = x^2 so x = +/- sqrt y

then
y^2 = 625
y = 25 or y = -25
x = sqrt 25 = +5 or -5
y = sqrt-25 = +5i or -5i

Any expression of the form a^2 - b^2 can be written as (a + b)(a - b).

Therefore, x^4 - 625) = (x^2 + 25)(x^2 - 25).

Then, (x^2 + 25)(x + 5)(x - 5)

To factor the expression x^4 - 625 completely, we can use the difference of squares formula. The formula states that the difference of two perfect squares, a^2 - b^2, can be factored as (a + b)(a - b).

In this case, we have x^4 - 625, which can be written as (x^2)^2 - 25^2. Now, we can see that a = x^2 and b = 25. Applying the difference of squares formula, we have:

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

Notice that (x^2 - 25) can be further factored using the difference of squares formula again:

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

Therefore, the expression x^4 - 625 can be completely factored as:

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

To factor the expression x^4 - 625 completely, we can use the difference of squares formula:

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

In this case, we can treat x^4 as a^2 and 625 as b^2. So, the expression can be rewritten as:

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

Now, we have two perfect squares: x^2 and 25. We can further factor the expression using the difference of squares formula:

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

Now, we have factored the expression completely. The final factorization is:

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

Note that (x^2 - 25) can be factored further as a difference of squares:

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

So, alternatively, the factorization can be written as:

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