p307 q10
prove that
$\displaystyle x^5 -1 = (x-1)(x^2-2x cos \frac {2\pi}{5} +1)(x^2-2xcos\frac{4\pi}5+1)$
would anyone just give me some hints on this type of question? thanks.
Hello,
$\displaystyle x^5-1=(x-1)(1+x+x^2+x^3+x^4)$
A polynomial of degree 4 is necessarily a product of two polynomials of degree 2.
Why ?
This has to do with the d'Alembert-Gauss theorem, which states that any polynomial can be developped into a product of polynomial of degree 1 containing complex numbers.
However, I don't know (or don't remember I've learnt it..) why a polynomial can always be a product of polynomials of degree 1 or 2...
Getting back to the problem :
$\displaystyle x^4+x^3+x^2+x+1=(x^2+ax+b)(x^2+cx+d)$
Develop in order to find a, b, c and d...
Edit : it seems that the first attempts are vain for solving..
The attachment I gave at this thread: http://www.mathhelpforum.com/math-he...-hard-way.html
might be of indirect interest to you.