Question

When (x^9 - x) is factored as completely as possible into polynomials and monomials with integral coefficients, how many factors are there?

Answer 1

Notice that

$(x^9-x)=x(x^8-1)$

Furthermore,

$(x^8-1)=(x^4+1)(x^4-1)=(x^4+1)(x^2+1)(x^2-1)=(x^4+1)(x^2+1)(x+1)(x-1)$

Then,

$(x^9-x)=x(x^4+1)(x^2+1)(x+1)(x-1)$

This expression cannot be further simplified. There are 5 factors in total, one monomial and four binomials.