Algebra

Algebra Curriculum

This curriculum provides a structured overview of the study of algebra, from elementary to abstract.

Elementary Algebra

Basic Operations:

• Review of arithmetic operations (addition, subtraction, multiplication, division)
• Order of operations (PEMDAS/BODMAS)
• Properties of operations (commutative, associative, distributive)
• Working with signed numbers (positive and negative numbers)
• Introduction to variables and expressions

Linear Equations and Inequalities:

• Solving one-variable linear equations
• Solving one-variable linear inequalities
• Graphing linear equations and inequalities on a number line
• Solving systems of two linear equations (substitution, elimination)
• Applications of linear equations and inequalities

Introduction to Polynomials:

• Identifying and classifying polynomials (monomials, binomials, trinomials)
• Adding and subtracting polynomials
• Multiplying polynomials (including FOIL)
• Dividing polynomials (long division, synthetic division)
• Factoring polynomials (common factors, difference of squares, trinomials)

Basic Algebra

Functions and Graphs:

• Definition of a function
• Function notation (e.g., f(x))
• Domain and range of a function
• Graphing functions (linear, quadratic, etc.)
• Transformations of functions (shifts, stretches, reflections)

Quadratic Equations:

• Solving quadratic equations by factoring
• Solving quadratic equations using the square root property
• Solving quadratic equations using the quadratic formula
• Graphing quadratic functions (parabolas)
• Applications of quadratic equations

Exponents and Radicals:

• Properties of integer exponents
• Simplifying expressions with exponents
• Scientific notation
• Definition of radicals and rational exponents
• Simplifying radical expressions

Further Algebra

Systems of Equations and Inequalities:

• Solving systems of linear equations in three or more variables
• Solving systems of linear inequalities
• Matrix operations (addition, subtraction, multiplication)
• Solving systems using matrices (Gaussian elimination, Cramer's Rule)
• Applications of systems of equations and inequalities

Polynomial Functions:

• Graphing polynomial functions
• Finding zeros of polynomial functions (rational root theorem, factor theorem)
• The Fundamental Theorem of Algebra
• Analyzing polynomial graphs (end behavior, multiplicity of zeros)
• Applications of polynomial functions

Rational Expressions and Equations:

• Simplifying rational expressions
• Adding, subtracting, multiplying, and dividing rational expressions
• Solving rational equations
• Applications of rational expressions and equations

Linear Algebra

Vectors and Vector Spaces:

• Definition of a vector and a vector space
• Examples of vector spaces (e.g., R^n, matrices, polynomials)
• Vector operations (addition, scalar multiplication)
• Subspaces and spanning sets
• Linear independence and bases

Matrices and Linear Transformations:

• Definition of a matrix
• Matrix operations (addition, multiplication)
• Matrix inverses and determinants
• Linear transformations and their matrix representations
• Eigenvalues and eigenvectors

Systems of Linear Equations:

• Solving systems using Gaussian elimination and Gauss-Jordan elimination
• Matrix form of a system of linear equations (Ax = b)
• Rank and nullity of a matrix
• Applications of linear systems
• LU Decomposition

Abstract Algebra

Groups:

• Definition of a group
• Examples of groups (cyclic groups, permutation groups, matrix groups)
• Subgroups and group homomorphisms
• Cosets and Lagrange's Theorem
• Quotient groups and isomorphism theorems

Rings:

• Definition of a ring
• Examples of rings (integers, polynomials, matrices)
• Ring homomorphisms and ideals
• Quotient rings and isomorphism theorems
• Integral domains and fields

Fields:

• Definition of a field
• Examples of fields (rational numbers, real numbers, complex numbers, finite fields)
• Field extensions
• Galois theory
• Applications of field theory