How to reduce x^4+4 into real factors

I am trying to factor a polynomial into irreducible factors in and got it to the following stage:

I noticed that wolfram has it reduced even further by splitting into . I was just wondering how this was done? I understand how to split the using complex numbers; however, nothing occured to me in

Also, just out of interest, if you have an equation using the variable , for example, and you want to talk about it within the set of real numbers, is it correct to say or should you say ?

Re: How to reduce x^4+4 into real factors

One way:

Re: How to reduce x^4+4 into real factors

Re: How to reduce x^4+4 into real factors

Quote:

Originally Posted by

**terrorsquid** I am trying to factor a polynomial into irreducible factors in

and got it to the following stage:

I noticed that wolfram has it reduced even further by splitting

into

. I was just wondering how this was done? I understand how to split the

using complex numbers; however, nothing occured to me in

Also, just out of interest, if you have an equation using the variable

, for example, and you want to talk about it within the set of real numbers, is it correct to say

or should you say

?

You should know that a real polynomial can be reduced to a product of real linear factors and real quadratic factors, then brute force will do the job:

gives the equations:

and the only real solutions to these equations gives the factorisation required.

CB

Re: How to reduce x^4+4 into real factors

Here's how I would have done it: . The two square roots of 2i are and . The two square roots of -2i are and so linear factors are

Since you want want **real** factors multiply and so that