Thread: Irreduclible polynomials

How can I show that these polynomails are irreducible over rationals.


f(x) = x^4+x-1
g(x)= x^4+8x+12
h(x) = 1+x+x^2/2+x^3/6+x^4/24
j(x) = x^4-4x-1

Originally Posted by ZetaX

f(x) = x^4+x-1

Write where and show this is impossible by comparing coefficients.

You can try similar arguments for the others.

Hello, ZetaX!

I would assume you're expected to use the Rational Roots Theorem.

Given a polynomial , if , then is a factor of

How can I show that these polynomials are irreducible over rationals?

All the polynomials have a leading coefficient of 1.
. . So the only possible rational roots are factors of the constant term.

For (1), the only choices are: . Try them in

For (2), the only choices are: .

For (3), the only choices are: .

For (4), the only choices are: .

Originally Posted by Soroban

Hello, ZetaX!

I would assume you're expected to use the Rational Roots Theorem.

Given a polynomial , if , then is a factor of


All the polynomials have a leading coefficient of 1.
. . So the only possible rational roots are factors of the constant term.

For (1), the only choices are: . Try them in

For (2), the only choices are: .

For (3), the only choices are: .

For (4), the only choices are: .

Be careful if a fourth degree polynomial has no zeros does not mean it is irreducible.
Consider, , no rational zeros but it can be factored.