Algebra

What is Algebra?

Algebra is one of the broadest and most fundamental branches of mathematics. At its core, algebra is about finding unknown values by using letters (called variables) to represent numbers in equations and formulas. The word "algebra" comes from the Arabic word al-jabr, meaning "reunion of broken parts," from the title of a 9th-century book by mathematician al-Khwarizmi.

Algebra provides the language and tools that are essential to nearly every area of mathematics, science, engineering, economics, and computer science. When you learn algebra, you're not just learning to solve equations — you're learning to think logically and abstractly.

Variables and Expressions

A variable is a symbol (usually a letter like x, y, or z) that represents an unknown or changeable value. An algebraic expression is a combination of variables, numbers, and operations.

Key Terminology

• Constant: A fixed value, such as 5, -3, or π
• Variable: A symbol representing an unknown value, like x or y
• Coefficient: The number multiplied by a variable, e.g., in 7x, the coefficient is 7
• Term: A single number, variable, or product of numbers and variables (e.g., 3x², -5y, 12)
• Expression: A combination of terms connected by + or - signs (e.g., 3x² + 2x - 5)

Order of Operations (PEMDAS)

When evaluating expressions, follow this order:

1. Parentheses — evaluate expressions inside parentheses first
2. Exponents — evaluate powers and roots
3. Multiplication and Division — from left to right
4. Addition and Subtraction — from left to right

Linear Equations

A linear equation is an equation where the highest power of the variable is 1. The graph of a linear equation is always a straight line.

Standard Form

Where a, b, and c are constants, and a ≠ 0.

Solving Linear Equations

The goal is to isolate the variable on one side of the equation using inverse operations:

Slope-Intercept Form

Where m is the slope (rate of change) and b is the y-intercept (where the line crosses the y-axis).

• The slope m = (y₂ - y₁) / (x₂ - x₁) measures steepness
• Positive slope: line goes up from left to right
• Negative slope: line goes down from left to right
• Zero slope: horizontal line
• Undefined slope: vertical line

Inequalities

An inequality compares two expressions using symbols like <, >, ≤, or ≥. Unlike equations, inequalities have a range of solutions.

Solving Inequalities

Solve inequalities the same way as equations, with one critical rule:

Compound Inequalities

A compound inequality combines two inequalities joined by "and" or "or":

• "And" inequality: Both conditions must be true (intersection). Example: -3 < x < 5
• "Or" inequality: At least one condition must be true (union). Example: x < -2 or x > 4

Polynomials

A polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication, with non-negative integer exponents.

Classification by Degree

• Constant: degree 0 (e.g., 7)
• Linear: degree 1 (e.g., 3x + 2)
• Quadratic: degree 2 (e.g., x² - 4x + 3)
• Cubic: degree 3 (e.g., 2x³ + x² - 5x + 1)
• Quartic: degree 4
• Quintic: degree 5

Polynomial Operations

Addition/Subtraction: Combine like terms (same variable and exponent).

Multiplication: Use the distributive property (FOIL for binomials).

Quadratic Equations

A quadratic equation has the standard form:

Methods for Solving Quadratics

1. Factoring

If the quadratic can be written as a product of two binomials, set each factor equal to zero.

2. Quadratic Formula

This formula works for any quadratic equation. The discriminant Δ = b² - 4ac determines the nature of the roots:

• Δ > 0: Two distinct real roots
• Δ = 0: One repeated real root
• Δ < 0: Two complex conjugate roots

3. Completing the Square

Transform the equation into the form (x + p)² = q, then take the square root of both sides.

The Vertex Form

The vertex of the parabola is at the point (h, k). If a > 0, the parabola opens upward; if a < 0, it opens downward.

Functions

A function is a rule that assigns to each input exactly one output. We write f(x) to denote the output of function f when the input is x.

Function Notation

If f(x) = 2x + 3, then:

• f(0) = 2(0) + 3 = 3
• f(4) = 2(4) + 3 = 11
• f(-1) = 2(-1) + 3 = 1

Domain and Range

• Domain: The set of all possible input values (x-values)
• Range: The set of all possible output values (y-values)

Types of Functions

• Linear function: f(x) = mx + b (straight line)
• Quadratic function: f(x) = ax² + bx + c (parabola)
• Absolute value function: f(x) = |x| (V-shape)
• Exponential function: f(x) = aˣ (rapid growth or decay)
• Logarithmic function: f(x) = log(x) (inverse of exponential)
• Polynomial function: f(x) = aₙxⁿ + ... + a₁x + a₀
• Rational function: f(x) = p(x)/q(x) where p and q are polynomials

Function Composition

First apply g to x, then apply f to the result.

Systems of Equations

A system of equations is a set of two or more equations with the same variables. The solution is the set of values that satisfies all equations simultaneously.

Methods of Solving

1. Substitution Method

Solve one equation for one variable, then substitute into the other equation.

2. Elimination Method

Add or subtract equations to eliminate one variable.

3. Graphing Method

Graph both equations and find the intersection point(s).

Types of Solutions

• One solution: Lines intersect at exactly one point (consistent and independent)
• No solution: Lines are parallel (inconsistent)
• Infinitely many solutions: Lines are the same (consistent and dependent)

Exponents and Radicals

Exponents represent repeated multiplication. Understanding the laws of exponents is crucial for simplifying expressions.

Laws of Exponents

• Product Rule: aᵐ · aⁿ = aᵐ⁺ⁿ
• Quotient Rule: aᵐ / aⁿ = aᵐ⁻ⁿ
• Power Rule: (aᵐ)ⁿ = aᵐⁿ
• Zero Exponent: a⁰ = 1 (when a ≠ 0)
• Negative Exponent: a⁻ⁿ = 1/aⁿ
• Fractional Exponent: a^(m/n) = ⁿ√(aᵐ)
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Radicals

A radical is the inverse operation of an exponent. The most common is the square root:

Simplifying Radicals

Factoring Techniques

Factoring is the process of writing an expression as a product of simpler expressions. It's essential for solving equations and simplifying rational expressions.

Common Factoring Techniques

1. Greatest Common Factor (GCF)

Factor out the largest factor common to all terms:

2. Difference of Squares

3. Perfect Square Trinomials

4. Sum and Difference of Cubes

5. Factoring by Grouping

Rational Expressions

A rational expression is a fraction whose numerator and denominator are both polynomials. Understanding rational expressions is essential for advanced algebra and calculus.

Simplifying Rational Expressions

Factor the numerator and denominator, then cancel common factors:

Operations with Rational Expressions

Multiplication

Factor first, cancel common factors, then multiply what remains.

Division

Multiply by the reciprocal of the divisor.

Addition and Subtraction

To add or subtract rational expressions, find a common denominator:

Solving Rational Equations

To solve an equation containing rational expressions, multiply both sides by the LCD to eliminate fractions, then solve the resulting polynomial equation. Always check for extraneous solutions — values that make the original denominators zero.

Logarithms

A logarithm answers the question: "What exponent do I need?" If bˣ = y, then log_b(y) = x. Logarithms are the inverse of exponential functions.

Common Logarithms

• Common log: log(x) = log₁₀(x) — used in science and engineering
• Natural log: ln(x) = logₑ(x) — used in calculus and advanced mathematics
• Binary log: log₂(x) — used in computer science

Properties of Logarithms

Solving Logarithmic Equations

Solving Exponential Equations

When the variable is in the exponent, use logarithms to bring it down:

Complex Numbers

When we encounter equations like x² + 1 = 0, there is no real number solution because no real number squared gives −1. To solve such equations, mathematicians introduced the imaginary unit:

A complex number has the form a + bi, where a is the real part and b is the imaginary part.

Operations with Complex Numbers

Addition and Subtraction

Multiplication

Use FOIL and remember that i² = −1.

Complex Conjugate

The conjugate of z = a + bi is z̄ = a − bi. Key property:

Division

Multiply numerator and denominator by the conjugate of the denominator:

The Complex Plane

Complex numbers can be plotted on the complex plane (Argand diagram), where the horizontal axis represents the real part and the vertical axis represents the imaginary part.

Modulus (Absolute Value)

This is the distance from the origin to the point (a, b).

Polar Form

Where r = |z| is the modulus and θ = arg(z) is the argument (angle).

Powers of i

The powers of i cycle with period 4:

• i¹ = i
• i² = −1
• i³ = −i
• i⁴ = 1
• i⁵ = i (cycle repeats)

To find iⁿ, divide n by 4 and use the remainder: iⁿ = i^(n mod 4).

Sequences and Series

A sequence is an ordered list of numbers following a pattern. A series is the sum of the terms of a sequence.

Arithmetic Sequences

Each term differs from the previous by a constant common difference d:

Geometric Sequences

Each term is multiplied by a constant common ratio r:

Sigma Notation

Useful summation formulas:

Absolute Value Equations and Inequalities

The absolute value of a number is its distance from zero on the number line. It is always non-negative:

Solving Absolute Value Equations

For |expression| = k where k > 0:

Solving Absolute Value Inequalities

Introduction to Matrices (Algebra Preview)

Matrices provide a powerful way to solve systems of equations and appear throughout higher mathematics. Here's a preview.

Representing Systems as Matrices

The system:

Can be written in matrix form as:

Solving with Cramer's Rule (2×2)

For the system ax + by = e and cx + dy = f:

This works when ad − bc ≠ 0 (the determinant is nonzero).

Algebraic Word Problems

Translating real-world situations into algebraic equations is one of the most practical skills in mathematics.

Strategy for Word Problems

1. Read the problem carefully — identify what is being asked
2. Define variables for the unknowns
3. Translate the conditions into equations
4. Solve the equations
5. Check that your answer makes sense in context

Common Problem Types