Factoring a polynomial

Algebra Level 5

We can treat polynomials with integer coefficients as polynomials modulo a prime, and factorize them modulo that prime. For example, the polynomial \( x^2 + x + 1 \) has no integer roots, but modulo 3 we can factorize it as \( x^2 + x + 1 = x^2 - 2x + 1 = (x-1)^2 \). In this case, we say that the polynomial is reducible modulo 3. On the other hand, this polynomial is not reducible modulo 2. Note that a reducible polynomial might have no roots, for example, \( x^4 + 2x^2 + 1 = (x^2 + 1)^2 \) is reducible in the integers, but it has no integer roots.

Which of the following is true for the polynomial \( x^4 - 10x^2 + 1 \)?

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