5 methods · 7 questions
When a question gives a polynomial through a property of its roots (a repeated root, all roots positive, roots in a given relation) and asks you to prove something about the coefficients (or vice versa), do not try to find the roots. Name them and write down the elementary symmetric relations , , equal to the (signed) coefficients, then push the given condition through those relations.
Trigger: a polynomial described by its roots, with the unknowns living in the coefficients.
Instances:
(i) For read off , , , then impose the extra condition.
(ii) A repeated root: set two named roots equal and eliminate to get a single relation between coefficients.
(iii) Symmetric functions of the roots (, ) rewritten purely in terms of the coefficients .
(iv) Auxiliary symmetric quantities (such as ) built from the roots to feed a later inequality.
Linked questions (1)