Let \(n \geq 1\) and let \(\Phi_n\) be the \(n\)th cyclotomic polynomial. Then \[ x^{n}-1=\prod_{\substack{d \divides n \\ d > 0}} \Phi_{d}(x), \qquad\text{so}\qquad \Phi_n(x) = \qty{x^n-1} \inverseof{\qty{\prod_{\substack{d \divides n \\ 0 < d < n}} \Phi_{d}(x)}}, \] and \[ \Phi_{n}(x)=\prod_{\substack{ d \divides n \\ d > 0} }\left(x^{d}-1\right)^{\mu\left(n/d\right)}, \] where \(\mu\) is the Möbius function: \[ \mu(m) = \begin{cases} 1 & \text{if } m = 1, \\ (-1)^{k} & \text{if } m \text{ is a product of } k \text{ distinct primes}, \\ 0 & \text{if } p^2 \divides m \text{ for some prime } p. \end{cases} \]
The first formula computes \(\Phi_n\) recursively: divide \(x^n - 1\) by the product of the \(\Phi_d\) with \(d \divides n\), \(d < n\), using polynomial long division.