Introduction to Algebra

• Algebra uses symbols to represent numbers and quantities.
• Algebra provides a framework for generalizing arithmetic operations and solving equations.

Basic Concepts

• A variable is a symbol that can represent an unknown number or a changing quantity.
• Constants are fixed values.
• An algebraic expression combines variables, constants, and mathematical operations (+, -, ×, ÷).
• An equation shows the equality between two expressions.
• Terms are the individual components of an algebraic expression, separated by + or - signs.
• Coefficients are the numerical part of a term that multiplies the variable.
• Like terms have the same variables raised to the same powers and can be combined.
• Unlike terms have different variables or different powers and cannot be combined.
• An operator indicates a mathematical operation like addition, subtraction, multiplication, division, or exponentiation.

Algebraic Expressions

• A monomial is an algebraic expression with one term.
• A binomial is an algebraic expression with two terms.
• A trinomial is an algebraic expression with three terms.
• A polynomial consists of one or more terms.
• The degree of a term is the sum of the exponents of its variables.
• The degree of a polynomial is the highest degree of its terms.

Simplifying Expressions

• Combining like terms involves adding or subtracting their coefficients.
• Distributive property: a(b + c) = ab + ac
• Expanding expressions involves multiplying out terms to remove parentheses.
• Factoring breaks down an expression into simpler terms.

Equations

• A linear equation has a highest variable power of 1.
• A quadratic equation has a highest variable power of 2.
• A system of equations includes two or more equations with the same variables.

Solving Linear Equations

• Isolate the variable to solve, maintaining equality on both sides.
• Use addition, subtraction, multiplication, or division to isolate the variable.
• Check the solution by substituting the value back into the original equation.

Solving Quadratic Equations

• Factoring involves setting the equation to zero, factoring, and solving for roots.
• Quadratic formula: For ax^2 + bx + c = 0, x = (-b ± √(b^2 - 4ac)) / (2a).
• Completing the square involves manipulating the equation to form a perfect square trinomial.

Systems of Linear Equations

• Substitution solves one equation for a variable and substitutes into the other.
• Elimination adds or subtracts equations to eliminate a variable.
• Graphing finds the intersection point of both equations to find the solution.
• Matrix methods use matrices and operations to solve the system.

Inequalities

• An inequality compares two expressions using symbols like <, >, ≤, or ≥.
• Solving inequalities is similar to solving equations.
• When multiplying or dividing by a negative number, reverse the inequality sign.

Functions

• A function relates inputs to outputs where each input has exactly one output.
• The input is the argument, and the output is the value of the function.
• Functions are represented by equations, graphs, or tables.
• Common functions include linear, quadratic, exponential, and trigonometric.

Graphing

• The coordinate plane is formed by the x-axis and y-axis.
• Points are represented by ordered pairs (x, y).
• Graphing involves plotting points that satisfy an equation and connecting them.
• The slope of a line measures its steepness and direction.
• The y-intercept is where the line crosses the y-axis.

Exponents and Radicals

• Exponents indicate repeated multiplication.
• Radicals are the inverse of exponentiation.
• Rules of exponents:
  • a^m * a^n = a^(m+n)
  • (a^m)^n = a^(m*n)
  • a^m / a^n = a^(m-n)
  • a^0 = 1
  • a^(-n) = 1/a^n

Logarithms

• A logarithm is the inverse of exponentiation.
• logarithm base b of x is the exponent to which b must be raised to equal x.
• Properties of logarithms:
  • log_b(xy) = log_b(x) + log_b(y)
  • log_b(x/y) = log_b(x) - log_b(y)
  • log_b(x^n) = n*log_b(x)

Polynomial Operations

• Adding and subtracting polynomials involves combining like terms.
• Multiplying polynomials uses the distributive property.
• Dividing polynomials uses long or synthetic division.

Factoring Techniques

• Greatest Common Factor (GCF) involves factoring out the largest common factor.
• Difference of Squares: a^2 - b^2 = (a + b)(a - b)
• Perfect Square Trinomial: a^2 + 2ab + b^2 = (a + b)^2 or a^2 - 2ab + b^2 = (a - b)^2
• Factoring by Grouping involves grouping terms and factoring out the GCF.

Rational Expressions

• Can simplify algebraic fractions.
• Operations with rational expressions:
  • Adding or subtracting requires a common denominator.
  • Multiplying involves multiplying numerators and denominators.
  • Dividing involves multiplying by the reciprocal of the divisor.

Absolute Value

• The absolute value is the distance from zero.
• Absolute value equations and inequalities can be solved by considering positive and negative cases.

Word Problems

• Translate word problems into algebraic equations or inequalities.
• Define variables to represent unknown quantities.
• Solve the equations or inequalities.
• Check the solution in the context of the problem.