Algebra Review

Precise definition

Algebra studies quantities and relationships using symbols under defined operations. An expression has a value; an equation asserts two expressions are equal for specified variable values; an identity is true throughout a domain. Solving means finding the complete solution set while preserving equivalence or checking any one-way transformations.

Notation and mathematical language

Key language includes coefficient, term, factor, exponent, root, domain, function, and inverse operation. The distributive law $a(b+c)=ab+ac$ and exponent laws such as $a^ma^n=a^{m+n}$ have conditions; for $a^m/a^n$, require $a\ne0$.

Conceptual picture

Algebraic fluency is structural: recognize common factors, balance equations, choose useful forms, and connect equations to graphs. Memorizing moves such as 'send it across' hides the operation applied to both sides and causes sign errors.

Fully worked example

Interpretation and application

Algebra supports geometry, calculus, probability, finance, and modelling. Different forms reveal different features: factored form exposes zeros, vertex form extrema, and standard form coefficients. Equivalent forms must retain the same domain.

Common mistakes