Algebra builds in layers, and Preliminaries is the first one: the exponent rules, radicals, polynomial arithmetic, factoring and number-system extensions that almost every later topic quietly assumes you already have. All 20 lessons in this chapter are published, each with a properties table, worked examples, a section on the mistakes that cost the most marks, and practice problems with hidden step-by-step answers — plus a matching practice set for unlimited drilling once the lesson itself makes sense.
They are deliberately ordered, not just grouped. Each lesson leans on the ones above it and nothing below it, so when a step looks like it came from nowhere, the explanation is usually one lesson earlier.
Exponents
The rules that everything else is written in. Start here even if they feel familiar — the later factoring and radical lessons assume you can move an exponent around without thinking about it.
- Integer Exponents — the product, quotient, power, zero and negative rules, derived from repeated multiplication rather than handed to you. · Practice
- Negative Exponents — what a negative exponent actually means, and why the result is never a negative number. · Practice
- Exponent Rules — every rule on one reference page, plus a strategy for expressions that need several of them at once. · Practice
- Rational Exponents — what
means, and why every integer-exponent rule still applies unchanged once the exponent is a fraction. · Practice
Radicals
The same ideas as the exponent lessons, written with a root sign instead.
- Radicals — the product and quotient properties, combining like radicals, and multiplying them. · Practice
- Simplifying Radicals — extracting perfect-power factors, the step most often left half-finished. · Practice
- Rationalizing Denominators — clearing a root from the bottom of a fraction, including the conjugate method that reappears in complex division. · Practice
Polynomial Arithmetic
- Polynomials — degree, standard form, and the vocabulary the next several lessons use. · Practice
- Adding Polynomials — identifying like terms and combining coefficients, horizontally and vertically. · Practice
- Subtracting Polynomials — the same process with the one step that causes most errors: distributing the negative across every term. · Practice
- Multiplying Polynomials — monomial times polynomial, FOIL, the box method, and the special product patterns worth memorising outright. · Practice
Factoring
Multiplication run backwards. This is the group that later chapters lean on hardest — nearly every quadratic and inequality problem factors something.
- Factoring Polynomials — the overview: GCF, difference of squares, perfect square trinomials and the ac method. · Practice
- Greatest Common Factor — finding the GCF of numbers, monomials and polynomial terms, and pulling it out completely. Always the first move. · Practice
- Factoring by Grouping — four-term polynomials in pairs, and the same technique applied to a trinomial with a leading coefficient other than 1. · Practice
- Factoring Quadratics — every trinomial case in one place, including how to tell when a quadratic will not factor at all. · Practice
- Factoring Higher-Degree Polynomials — sum and difference of cubes, grouping on bigger polynomials, and quartics that are quadratic in form. · Practice
Rational Expressions
Fractions built from polynomials, which is why factoring had to come first.
- Rational Expressions — simplifying, multiplying, dividing, adding and subtracting them. · Practice
- Simplifying Rational Expressions — factor then cancel, and why an excluded value survives even after the factor that caused it disappears. · Practice
Complex Numbers
The chapter closes by extending the number system, and by reusing the conjugate trick from the radical lessons on a completely different problem.
- Complex Numbers — the imaginary unit, the repeating powers of
, and where complex answers come from. · Practice - Complex Number Arithmetic — adding, subtracting, multiplying with
, and dividing by multiplying through by the conjugate. · Practice
Where to Go From Here
Working through the chapter for the first time, start at the top and read down. If you already know a topic, skip the lesson and go straight to its practice set — clearing a round of Hard problems is a faster test of whether you actually know it than re-reading an explanation you half remember.
Once this chapter is behind you, Solving Equations and Inequalities is where all of it gets used: linear and quadratic equations, word problems, radical equations, inequalities and absolute value, with its own hub listing every lesson in order. The full Algebra chapter list and the complete set of math practice problems are worth a look too.