For each polynomial below determine whether the polynomial i

For each polynomial below, determine whether the polynomial is irreducible over Q or not. State your reasons, and show all your work. X^2 + x +4. 3x^7 - 10x^4 + 20. -x^3 - 4x^2 + 8.

Solution

If f(x) Z[x], then f(x) factors into a product of two polynomials of lower degrees r and s in Q[x] if and only if it has such a factorization with polynomials of the same degrees r and s in Z[x]. So if we want to determine whether a polynomial is irreducible in Q[x], it suffices to check whether it can be factored in Z[x].Further, as per the Rational Roots theorem, If f(x) = a_n xⁿ + · · · + a₀ is in Z[x] with a₀ 0, and if f(x) has a zero p/ q Q (written in lowest terms), then p | a₀ and q | a_n in Z.