Introduction to Algebra

• Algebra uses symbols to represent numbers and quantities, generalizing arithmetic operations.

Basic Concepts

• Variable: A symbol, often a letter, representing an unknown or changeable quantity.
• Constant: A fixed value that remains the same.
• Expression: A combination of variables, constants, and operations.
• Equation: A statement that shows the equality between two expressions.
• Coefficient: A number multiplied by a variable in an algebraic expression.
• Operator: Symbols indicating mathematical operations, like +, -, ×, ÷.
• Term: A single number or variable, or numbers and variables multiplied together, separated by + or - signs in an expression.

Key Operations

• Addition: Combining terms; only like terms can be added.
• Subtraction: Finding the difference between terms; only like terms can be subtracted.
• Multiplication: Multiplying coefficients and adding exponents of like variables.
• Division: Dividing coefficients and subtracting exponents of like variables.
• Exponents: Indicate repeated multiplication of a base; a^n means a multiplied by itself n times.

Order of Operations

• Parentheses should be resolved first.
• Exponents are calculated next.
• Multiplication and Division are performed from left to right.
• Addition and Subtraction are performed from left to right.
• Acronyms like PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) help in remembering the order.

Solving Equations

• Linear Equations: Equations where the highest power of the variable is 1 (e.g., ax + b = c).
  • Isolate the variable using inverse operations on both equation sides.
• Quadratic Equations: Equations where the highest power of the variable is 2 (e.g., ax^2 + bx + c = 0).
  • Factoring: Express the quadratic expression as a product of two binomials.
  • Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / (2a)
  • Completing the Square: Manipulating the equation to form a perfect square trinomial.

Simplifying Expressions

• Combining Like Terms: Add or subtract terms that have the same variable raised to the same power.
• Distributive Property: a(b + c) = ab + ac
• Factoring: Expressing an expression as a product of its factors.

Inequalities

• Inequalities use symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to) to show the relationship between expressions.
• Solving Inequalities: It's similar to solving equations, with the key difference that multiplying or dividing by a negative number reverses the inequality sign.
• Graphing Inequalities: Represent the solution set on a number line or coordinate plane.

Functions

• Functions relate a set of inputs to a set of permissible outputs, where each input is related to exactly one output.
• Notation: f(x) represents the output of the function f for the input x.
• Domain: The set of all possible input values (x-values) for which the function is defined.
• Range: The set of all possible output values (y-values) that the function can produce.
• Types of Functions: Linear, quadratic, polynomial, exponential, logarithmic, trigonometric.

Graphing

• Coordinate Plane: The x-axis and y-axis define a plane.
• Plotting Points: Points are represented as ordered pairs (x, y) on the coordinate plane.
• Linear Equations: Generally graph as straight lines, described by y = mx + b (m is slope, b is y-intercept).
• Slope: The steepness of a line, calculated as rise over run (change in y divided by change in x).
• Intercepts: The x-intercept is where the graph crosses the x-axis; the y-intercept is where the graph crosses the y-axis.

Systems of Equations

• These are sets of two or more equations with the same variables.
• Solving Methods:
  • Substitution: Solve for one variable in one equation and substitute that expression into the other equation.
  • Elimination: Add or subtract equations to eliminate one variable.
  • Graphing: Find the intersection point of the equations' graphs.

Polynomials

• Expressions with variables and coefficients, using addition, subtraction, multiplication, and non-negative integer exponents.
• General Form: a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0
• Degree: The highest exponent of the variable in the polynomial.
• Operations:
  • Addition and Subtraction: Combine like terms.
  • Multiplication: Use the distributive property to multiply each term in one polynomial by each term in the other polynomial.
• Factoring Polynomials: Breaking down a polynomial into simpler factors.

Radicals

• Radicals are mathematical expressions using roots, like square roots, cube roots, etc.
• Square Root: √a represents the non-negative number that, when squared, equals a.
• Simplifying Radicals: Simplify by removing any perfect square factors from under the radical sign.
• Operations with Radicals:
  • Addition and Subtraction: Combine like radicals (radicals with the same index and radicand).
  • Multiplication: Multiply coefficients and radicands separately.
  • Division: Simplify by rationalizing the denominator if needed.
• Rationalizing the Denominator: Eliminating radicals from the denominator of a fraction.

Rational Expressions

• Fractions where the numerator and/or the denominator are polynomials.
• Simplifying Rational Expressions: Factor the numerator and denominator and cancel out common factors.
• Operations with Rational Expressions:
  • Addition and Subtraction: Find a common denominator and combine the numerators.
  • Multiplication: Multiply the numerators and denominators separately.
  • Division: Invert the second fraction and multiply.

Word Problems

• Verbal information is translated into algebraic equations or expressions.
• Identify key information, define variables, and set up equations.
• Solve the equations and interpret the results in the context of the problem.

Key Algebraic Identities

• (a + b)^2 = a^2 + 2ab + b^2
• (a - b)^2 = a^2 - 2ab + b^2
• (a + b)(a - b) = a^2 - b^2
• (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
• (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3