How is the formula for expressing a cyclotomic polynomial derived.

Question

$\frac{x^p-1}{x-1}=x^{p-1}+x^{p-2}+.....+x+1=f(x)$ a cyclotomic polynomial.

But I cant seem to work out how this equality is found.

I Tried expressing it as a geometric series

$$\frac{x^p-1}{x-1}=\frac{x^p-1}{x(1-\frac{1}{x})}=\frac{x^p-1}{x}\frac{1}{1-\frac{1}{x}}=\frac{x^p-1}{x}\sum_{n=0}^p\frac{1}{x^n}$$

But this didn't produce the correct equation , does anyone have any suggestions ?

Answer 1

Hint Calculate $$(x-1)(x^{p-1}+x^{p-2}+...+x+1)$$ by doing the multiplication.

Answer 2

I am not sure about your terminology. But $$x^{n+1}-1=(x-1)(x^n+x^{n-1}+\cdots+x+1)$$ for any $n\in\Bbb{N}.$ For example $x^2-1=(x-1)(x+1)$ and $x^3-1=(x-1)(x^2+x+1)$ and so on.
One way to give a rigorous proof is for this identity is induction and there are so many other ways also.