Mastering Prime Factorisation of Polynomials

Prime factorisation is a method of expressing a polynomial as a product of its ______.

The Rational Roots Theorem can be helpful in determining the ______ of a polynomial.

The polynomial $x^{12}-1$ can be factored as $(x^6-1)(x^6+1)$. Further decomposition of $x^6-1$ is left to ______.

The polynomial $x^6+1$ can be factored as $(x^2+1)(x^4-x^2+1)$. We would like to show that $x^4-x^2+1$ is ______.

To express $x^4-x^2+1$ as a product of quadratics with integer coefficients, the decomposition would have the shape $(x^2+ax+b)(x^2+cx+d)$, without loss of ______.