Math Is Fun Forum

The link doesn't work.

If you wish to express the roots as the sums of cosines, then you use a prime number of the form p=6m+1. With m as the "number of cosines" that you use. So for you, we have p=31=6m+1. Thus, m=5. Meaning the roots are

Note that this doesn't work for any cubic as the constant has to be a specific term in order to be able to express the roots as the pth roots of unity.

So you're saying that to solve this problem, first pretend that Gina and Christina are the same people, and then find a way to arrange 7! people?

Okay, can I say minimize? Like, if computation by hand is possible, then you do it by hand?

I just want to stay away from computers because I think that they don't really improve your mental math skills and encourage laziness! (For me at least)

According to Wikipedia, the roots are equal to

So the rational root theorem won't work.

But I'm wondering if it's possible to compute it mostly by hand. Like, not get a computer software, plug in the polynomial and have the roots shown instantly.

I think this may have something to do with Galois Theory... but I'm not too sure.

I am getting

as one of the roots, is that right?? I don't trust my math.

@Zetafunc

Actually, the sextic can be factored into

Now, I just need to find the roots of the cubics...

Going through my old threads, I have decided to answer this question, now that I know how to simplify.

In general, the roots of any cubic equation

can also be written into the form

where

are the roots of the cubic,

and d is the negated discriminant. More specifically,

Actually, I wonder if it's possible to factor it into two cubics, and solving for the roots of the cubics generate the roots of the six degree... hm...

@Zetafunc

Does the polynomial I asked have a solvable galois group? And also, what's a galois group. I haven't gotten that far in math yet.

Yes. That's supposed to be there.

The midpoint formula is

where

is one endpoint, and

is the other.

Since we know one endpoint and know the midpoint, using the midpoint formula, we can find the unknown endpoint.

So plugging everything in, we obtain

For x, we multiply both sides by 2 and subtract 3 to get

. For y, we (again) multiply both sides by 2 and subtract 5 to get

. Therefore,