PhD Qualifying Exam

Mathematics Preparation Series

About This Site

This is a comprehensive collection of mathematics lessons designed for PhD qualifying exam preparation. Study on the go from your phone or any device with beautifully formatted, mobile-responsive lessons.

Lessons

Module M1: Foundations

Lesson M1.1: Real Numbers, Properties, and Order

Module M1
~2 hours
Prerequisites: None

Identify and classify numbers within the real number system, apply algebraic properties, work fluently with inequalities and absolute value, and understand the completeness property.

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Lesson M1.2: Algebraic Manipulation and Factoring

Module M1
~3 hours
Prerequisites: M1.1

Perform polynomial arithmetic fluently, apply all standard factoring techniques systematically, simplify rational expressions and complex fractions, and rationalize denominators.

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Lesson M1.3: Equations and Inequalities

Module M1
~4 hours
Prerequisites: M1.1, M1.2

Solve polynomial equations using factoring and the quadratic formula, solve rational and radical equations while tracking domain restrictions, and use sign analysis for rational inequalities.

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