Problem 1 – Fall 2020 – Symmetric Functions I

Symmetric Functions I

This problem is from Alexander Karabegov.

Given a quadratic polynomial x² + p x + q with two real roots s and t, express the following functions of s and t as polynomials in p and q.

1. s + t
2. st
3. s² + t²
4. s³ + t³

Please submit all problem solutions to mathpotw@acu.edu before 5:00 PM on the next Thursday. 

Solution to Symmetric Functions I

Correct solutions were submitted by:  Ricky Kagoro Lumala, Wyatt Witemeyer

Since s and t are roots of the polynomial x² + p x + q, then

x² + p x + q = (x–s)(x–t).Multiplication of the right side yields

x² + p x + q = x² – (s+t)x + s t.Thus the coefficients of the x term give the first result: s + t = –p.

The constant terms produce the second: st = q.

The third part can be obtained by observing that s² + t² = (s+t)² – 2st, and using the two previous results.
Therefore, s² + t² = (-p)² – 2q = p² – 2q.

Similarly, s³ + t³
= (s+t)³ – 3s²t – 3st²
= (s+t)³ – 3st(s+t)
= (-p)³ – 3q(-p)
= –p³ + 3p q.