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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