Algebra Portal

1Overview

Algebra expresses relations among numbers and functions by formulas and analyzes them through equivalent transformations, factorization, equations, and inequalities. This portal preserves the _order.json sequence while grouping related lectures into five paths.

2Polynomial path

Define the degree, coefficients, and roots of a polynomial; proceed through the remainder and factor theorems to factorization. Then introduce homogeneous expressions, whose terms have equal total degree.

3Quadratics and inequalities

Derive the quadratic formula by completing the square, and connect it to graphs of quadratic functions and inequalities. Then study inequality proofs using the arithmetic-geometric mean inequality and homogeneity.

4Exponentials, logarithms, and complex numbers

Organize exponential and logarithmic functions as inverse functions. After extending the number system from real to complex numbers, use the fundamental theorem of algebra to study the existence of roots of complex polynomials.

5Symmetry of expressions and functional equations

Distinguish symmetric and alternating expressions by invariance or sign change under permutations of variables. Functional equations are equations whose unknown is a function rather than a number.

6Connection to number theory

After introducing divisibility and congruences, proceed to the Euclidean algorithm and linear Diophantine equations, prime factorization, and then general Diophantine equations and infinite descent. The number theory portal covers subsequent topics such as the Chinese remainder theorem.