Problem
ALG-B1-M04-P021 Roots of a Mirrored Cubic
#21
★★★★★ Level 5 of 5
Distinct real numbers \(a_1,a_2,a_3\) and a number \(t\) are such that \((x-a_1)(x-a_2)(x-a_3)=t\) has three real roots \(c_1,c_2,c_3\). Find the roots of \((x+c_1)(x+c_2)(x+c_3)=t\).
Inspired by regional olympiad method · 2023 · Grade 11 · Problem 2
Write the identity of two monic cubic polynomials and substitute \(-x\).
The polynomial \((x-a_1)(x-a_2)(x-a_3)-t\) has roots \(c_1,c_2,c_3\), so it equals \((x-c_1)(x-c_2)(x-c_3)\).
Substitute \(-x\): \((-x-a_1)(-x-a_2)(-x-a_3)-t=(-x-c_1)(-x-c_2)(-x-c_3)\).
Multiplying by \(-1\), we obtain \((x+a_1)(x+a_2)(x+a_3)+t=(x+c_1)(x+c_2)(x+c_3)\). Therefore at \(x=-a_1,-a_2,-a_3\), the left side of the equation equals \(t\). Answer: \(-a_1,-a_2,-a_3\).
A strong but clean problem on identity of monic cubics.