7.2-03 Real and complex roots

Real and complex roots

Count real and complex eigenvalues carefully, separating distinct roots from algebraic multiplicity. Use polynomial factors, conjugate pairs and discriminants to recognize possible and impossible spectra.

What you will practise

• Count roots with multiplicity
• Solve simple complex quadratic equations
• Apply real-coefficient conjugate pairing
• Analyze a parameter using the discriminant

Prerequisites and study approach

This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review quadratic equations and complex arithmetic. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently.

The final spectrum check requires both dimension counting and the conjugate-pair restriction. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.