Symmetric Polynomials

Abstract

Let’s call the roots p, q and r. Then (x− p)(x− q)(x− r) = x + ax + bx + c so that p + q + r = −a pq + qr + rp = b pqr = −c Now the cubic we want is (x− p)(x− q)(x− r) = x − (p + q + r)x + (pq + qr + rp)x− pqr so our task is to express the symmetric polynomials p + q + r, pq + qr + rp and pqr in terms of the elementary symmetric polynomials s1 = p + q + r, s2 = pq + qr + rp and s3 = pqr. Of course, p qr = (pqr) = (−c) = −c, but how about the other two?