Thread: Finding the cosine of 72* the hard way

Alright, so let's say that w = cis(72*)

Using demoivre's theorem, w^5 = 1

What I'm hung up on, now, is showing that w^4+w^3+w^2+w+1=0, I'm supposed to use something I already know about polynomials (synthetic division?), and knowing that both w and 1 are roots of f(x) = x^5 -1, but I'm drawing a total blank.

All help is very much appreciated.

Originally Posted by Kwerk

Alright, so let's say that w = cis(72*)

Using demoivre's theorem, w^5 = 1

What I'm hung up on, now, is showing that w^4+w^3+w^2+w+1=0, I'm supposed to use something I already know about polynomials (synthetic division?), and knowing that both w and 1 are roots of f(x) = x^5 -1, but I'm drawing a total blank.

All help is very much appreciated.

Okay I get that, now I have to turn that into w^2 + w^3 = (w + w^4)^2 - 2
somehow. I think I'm supposed to expand (w+w^4) ^2 into w^8+2w^5+w^2, but after than I'm lost again.

Originally Posted by Kwerk

Alright, so let's say that w = cis(72*)

Using demoivre's theorem, w^5 = 1

What I'm hung up on, now, is showing that w^4+w^3+w^2+w+1=0, I'm supposed to use something I already know about polynomials (synthetic division?), and knowing that both w and 1 are roots of f(x) = x^5 -1, but I'm drawing a total blank.

All help is very much appreciated.

Read the attachment I have (quickly) prepared for you and work your way through it.