7 methods · 26 questions
To find how many real roots has (or to fix a parameter so the count is a target value), differentiate, locate the stationary points, and evaluate there. The number of times the graph crosses the axis is dictated by how that ordered list of stationary values straddles , together with the end behaviour. A turning value and the next one force a crossing between them; consecutive stationary values of the same sign give no crossing there. For a cubic this collapses to the clean rule: three distinct real roots exactly when local max and local min .
Trigger: a polynomial (or polynomial-after-rearranging) and a question about the number of real roots or the values of a constant giving a prescribed number.
Instances:
(i) evaluate at every stationary point and count sign changes across the list to read off the crossings;
(ii) for three distinct roots impose (local max value) (local min value) , which turns into a band on the parameter;
(iii) fix where the roots sit (e.g. how many are positive) by also testing the sign of at or another reference point.
Linked questions (5)