Course Description

This syllabus organizes the core topics of a typical Algebra 1 course — from set theory and the properties of real numbers through factoring, rational expressions, radicals, quadratic equations, and complex numbers. Each unit lists the learning objective and the specific skills/topics covered, matching the structure of the reference outline.

Unit Overview

Detailed Unit Breakdown

Unit 1: Set Theory & Foundations

Objective: Understand set notation, membership, subsets, and set operations (union, intersection, complement).

• Set notation: braces, elements, ellipses for patterns
• Membership symbols: ∈, ∉, and set relation symbols ⊂, ⊆, ⊄
• Empty set (∅) and set equality
• Union (∪), intersection (∩), and complement (Ā)
• Cardinality n(A) and equivalent sets (~)
• Number sets: Natural, Whole, Integers, Rational, Irrational, Real, Imaginary, Complex

Unit 2: Properties of Real Numbers

Objective: Apply the closure, commutative, associative, identity, inverse, and distributive properties, along with properties of equality and inequality.

• Properties for addition and multiplication (closure, commutative, associative, identity, inverse)
• Distributive property: a(b + c) = ab + ac
• Properties of equality: reflexive, symmetric, transitive, addition, multiplication
• Properties of inequality: trichotomy, transitive, addition, multiplication (sign changes)

Unit 3: Operations on Real Numbers

Objective: Perform addition, subtraction, multiplication, and division with signed numbers using absolute value rules.

• Absolute value and distance from zero
• Addition rules for same and different signs
• Subtraction as 'add the opposite'
• Multiplication and division sign rules
• Double negative rule

Unit 4: Algebraic Terms & Expressions

Objective: Combine like terms and multiply polynomials using the distributive property, FOIL, and exponent rules.

• Combining (adding/subtracting) like terms
• Product rule for exponents; multiplying coefficients and variables
• Distributive property for polynomials (monomial × polynomial, binomial × binomial, binomial × trinomial)
• FOIL method for multiplying two binomials
• Special products: (a+b)², (a−b)², (a+b)(a−b)
• Exponent rules: power of a power, power of a product, power of a quotient, zero power, negative exponents
• Dividing terms using the quotient rule

Unit 5: Solving First-Degree Equations & Inequalities

Objective: Solve linear equations and inequalities in one variable and verify solutions.

• Clearing fractions using the Multiplication Property of Equality
• Simplifying each side (order of operations, distributive property, like terms)
• Isolating the variable with the Addition Property of Equality
• Solving for the variable with the Multiplication Property of Equality
• Checking solutions; conditional equations, identities, and inconsistent equations
• Solving inequalities: addition and multiplication properties (reversing the sign)

Unit 6: Order of Operations

Objective: Simplify numeric and algebraic expressions in the correct sequence.

• Enclosure symbols: parentheses, brackets, braces, and fraction bars
• Exponents and roots (left to right)
• Multiplication and division (left to right)
• Addition and subtraction (left to right)

Unit 7: Factoring

Objective: Factor polynomials fully by recognizing GCF, trinomials, binomials, perfect cubes, and grouping patterns.

• Step 1 — Factoring out the Greatest Common Factor (GCF)
• Step 2 — Categorizing by term count and degree
• Trinomials: x² + bx + c and ax² + bx + c (trial and error, a ≠ 1)
• Binomials: difference of squares, sum/difference of cubes
• Perfect squares and prime (unfactorable) binomials
• Perfect cubes (4-term expansion)
• Grouping: (2–2), (3–1), and (1–3) patterns

Unit 8: Rational Expressions

Objective: Simplify, add, subtract, multiply, and divide rational expressions, including complex fractions.

• Domain restrictions (denominator ≠ 0)
• Reducing to lowest terms by factoring
• Addition and subtraction with same/different denominators; least common denominator
• Multiplication of rational expressions
• Division (multiply by the reciprocal)
• Complex fractions: simplifying numerator/denominator method or common-denominator method

Unit 9: Synthetic Division

Objective: Divide a polynomial by a binomial of the form x − h using synthetic division.

• Writing the polynomial in descending order with placeholder zero coefficients
• Setting up the divisor and bringing down the leading coefficient
• Multiply-and-add process across all coefficients
• Interpreting the quotient and remainder

Unit 10: Roots & Radicals

Objective: Simplify radical expressions and solve equations containing radicals.

• Radical notation, index, and radicand; rational exponents
• Rules for multiplying and dividing radicals with the same index
• Simplifying radicals with one term (greatest root, variable exponents)
• Simplifying radicals with multiple terms and factoring
• Rationalizing denominators (monomial and binomial/conjugate)
• Radical equations: isolating and raising both sides to a power; checking for extraneous solutions

Unit 11: Quadratic Equations

Objective: Solve second-degree equations by factoring or the quadratic formula.

• Standard form ax² + bx + c = 0
• Zero-product property
• Solving by factoring
• The quadratic formula and the discriminant

Unit 12: Complex Numbers

Objective: Perform arithmetic with complex numbers in the form a + bi.

• Definition of i and i² = −1
• Addition and subtraction of complex numbers
• Multiplication using FOIL and simplifying i²
• Division by rationalizing with the conjugate