Problem
ALG-B1-M04-P016 Sum of Roots of Two Trinomials
#16
★★★★★ Level 5 of 5
Monic quadratic trinomials \(f\) and \(g\) each have two real roots. Suppose \(f(2)=g(5)\) and \(g(2)=f(5)\). Find the sum of all four roots.
Inspired by regional olympiad method · 2019 · Grade 9 · Problem 1
Consider \(h(x)=g(7-x)\).
The polynomial \(h(x)=g(7-x)\) is also monic quadratic. The conditions give \(h(2)=g(5)=f(2)\) and \(h(5)=g(2)=f(5)\).
The difference \(h-f\) has degree at most \(1\), but has two roots \(2\) and \(5\). Hence \(h=f\).
If \(r\) is a root of \(f\), then \(7-r\) is a root of \(g\). The roots form two pairs with sum \(7\), so the total sum is \(14\).
Rewrites the idea of reflecting roots about the midpoint of two points.