# Polynomial divisors

Algebra Level 4

How many divisors of the polynomial $$x^4 - 1$$ in $$\mathbb{Z}[x]$$ are there?

Definitions:

• $$\mathbb{Z}[x]$$ is the ring of polynomials with integer coefficients. That is, elements of $$\mathbb{Z}[x]$$ are polynomials with integer coefficients. $$1, x, 2x^2 + 3x - 4$$ are elements of $$\mathbb{Z}[x]$$; $$\frac{1}{2}, x^{-1}, 2^x$$ aren't.

• A polynomial $$Q(x)$$ is a divisor of another polynomial $$P(x)$$ in $$\mathbb{Z}[x]$$ if (and only if) there exists a third polynomial $$R(x)$$ such that $$P(x), Q(x), R(x) \in \mathbb{Z}[x]$$ and $$P(x) = Q(x)R(x)$$.

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