Thread: 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 ?

One way:

too easy, thanks!

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

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