Do you know that ?
Do you know that ?
You can use both of those in the proof.
Right. This is really getting on my nerves.
The question is:
Let be a polynomial where the coefficientsn are real and
i). Show that if is a complex root of f then it's conjugate is also a root of f.
Would this just be that the product of a complex number and it's conjugate produce a real number. Since this polynomial has real coefficients then there can't be a complex number on it's own (or some of the coefficients would be complex)?
ii) Suppose that the degree n is odd. Show that f must have at least one real root.
Would the solution to this be that a polynomial with a degree n has n roots. If the polynomial is odd then there would be pairs of complex numbers and their conjugates and one root left over. Since this root is on it's own, it must be real to avoid complex coefficients.
I do not know how to prove this though .
P.S: Sorry for two threads in one night, there's no gentle introduction on a maths course!
I know both those facts, so thanks for putting me on the right lines!
So would I prove part i) by:
I get that far (I managed to use the first thing you wrote but just for ). I could do the same for but it's hard to try and prove that and are equal to 0.
This is one of my ideas to prove it is equal to 0:
since we know and
we could find which we might be able to cancel and get somewhere. I could use the second thing you wrote if I did it this way. If the sum of them both does come out to 0 since and (ie. the sign of the real part is the same) then . The only possible values of and are therefore 0.
Since the function is the same they are both roots of the function.
However, it now becomes a challenge to prove that . The complex parts would cancel out giving a sequence like:
(i would put in more but I have to dash off to my test in a minute).
Since I would end up with this, could equal 0. Unfortunately, I get this far and i'm not really sure what to do!
Is this the right method? It makes sense to me so far but i'm not gifted with on overabundance of logic!
P.S: I'm sorry for such a long post, ideas kept coming to me as I kept writing!