Thread: Discriminant question

Using the Theorem on Symmetric Functions or otherwise,
show that the discriminant of the quartic polynomial x^4 + qx + r
is 27q^4 + 256r^3.

Now, using the Symmetric Functions theorem, it's obvious that q= -s3 and r=s4 (where s3 and s4 are the third and fourth elementary symmetric polynomials). How do I proceed from here?

Many thanks.

Originally Posted by KSM08

Using the Theorem on Symmetric Functions or otherwise,
show that the discriminant of the quartic polynomial x^4 + qx + r
is 27q^4 + 256r^3.

Now, using the Symmetric Functions theorem, it's obvious that q= -s3 and r=s4 (where s3 and s4 are the third and fourth elementary symmetric polynomials). How do I proceed from here?

The discriminant is given by , where are the roots of the polynomial (over the complex field, say).

For a quartic polynomial, the discriminant is a symmetric function of the roots, of degree 12. But q has degree 3 and r has degree 4. The symmetric function theorem says that must be a function of the elementary symmetric functions, and in this example the only available such functions are q and r (because and are 0). So must be of the form , for some constants m and n.

To find m and n, evaluate in some special cases. For example, if q=0 and r=–1 then the equation becomes x^4=1, with roots . From the definition of , you can check that in this case. So n=256. Similarly, if q = –1 and r=0 then the equation is , with roots . I haven't tried to check this, but presumably works out as 27 in this case.