What is the answer to the expression when factoring it completely over the complex numbers?

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1 Answer
Oct 24, 2017

#x^4 - 625 = (x + 5)(x - 5)(x + 5i)(x - 5i)#

Explanation:

Knowing that #5^4 = 625#, we can immediately factor as

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

Another difference of squares appears.

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

We can factor #x^2 + a^2# as #(x + ai)(x - ai)#, where #i# is the imaginary unit. We do this because #x^2 + 25# cannot be factored on the real numbers.

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

If we try expanding #(x + 5i)(x - 5i) = x^2 - 25(i^2)# and #i^2 = -1#.

So, #(x + 5i)(x - 5i) = x^2 + 25#.

Hopefully this helps!