Classifcation of irreducible polynomials

[gazzo]

"Conjecture a classifucation rule for all irreducible polynomials of the form ax^2 + bx + c over the reals. Prove it."

I'm stuck cold at the start. classification rule ?

"Let R be an integral domain.
A nonzero f in R[x] is irreducible provided f is not a unit and in every factorization f = gh, either g or h is a unit in R[x].

So, f in R[x] is reducible over R if it can be factorised as f = gh where g,h are in R[x] with deg(g) < deg(f) and deg(h) < deg(f). And irreducible otherwise."

I have no idea where to start, I tried playing with extensions but that seems pointless in the reals.

For some p, b^2 - 4ac < 0 then p is irreducible over R. But that's not getting me anywhere.

Could someone please just give me a tiny hint/word which may shed a ray of light?

Thanks

 

[Nate810]

The classification rule for all irreducible polynomials of the form ax^2 + bx + c over the reals is that any polynomial where b^2 - 4ac < 0 is irreducible over R. This can be proven by showing that any polynomial with b^2 - 4ac greater than or equal to 0 can be factored using the quadratic formula.

 

[quicknote]

One possible classification rule for irreducible polynomials of the form ax^2 + bx + c over the reals is to consider the discriminant, b^2 - 4ac. If this value is negative, then the polynomial is irreducible over the reals. This is because the discriminant indicates the nature of the roots of the polynomial - if it is negative, then the roots are complex and cannot be expressed as real numbers. Therefore, the polynomial cannot be factored into linear factors over the reals.

To prove this, we can use the quadratic formula to find the roots of the polynomial. If the discriminant is negative, then the square root in the formula will be imaginary, resulting in complex roots. This means that the polynomial cannot be factored into linear factors over the reals, making it irreducible.

On the other hand, if the discriminant is non-negative, then the roots will be real and the polynomial can be factored into linear factors over the reals. Therefore, the polynomial is reducible if the discriminant is non-negative.

In summary, a polynomial of the form ax^2 + bx + c is irreducible over the reals if and only if the discriminant, b^2 - 4ac, is negative. This serves as a classification rule for all irreducible polynomials of this form over the reals.