Introduction to Algebra

• Algebra involves the manipulation of symbols and the rules governing them.
• Symbols represent quantities without a defined value, serving as variables.
• Algebra is interconnected with number theory, geometry, and analysis.

Variables and Expressions

• A variable is a symbol, often a letter, that stands in for an unknown.
• An expression combines variables, numbers, and operational signs (+, -, ×, ÷).
• 3x + 5y - 2, exemplifies what an expression looks like.

Equations

• An equation shows that two expressions are equal.
• The equals sign (=) is a necessary symbol in equations.
• 3x + 5 = 14 serves as an example of an equation.

Solving Equations

• Solving equations involves determining the value(s) of the variable(s) that satisfy the equation.
• To solve an equation, isolate the variable on one side.
• Isolate variables by using inverse operations like addition/subtraction or multiplication/division, to maintain equality.

Linear Equations

• A linear equation's variable has a highest power of 1.
• A linear equation can be expressed as ax + b = c, with a, b, and c representing constants.
• Solving linear equations depends upon isolating the variable via algebraic operations.

Quadratic Equations

• A quadratic equation is characterized by a variable with a maximum power of 2.
• The standard format for quadratic equations is ax² + bx + c = 0, where a ≠ 0.
• Factoring, completing the square, or using the quadratic formula are common ways to solve quadratic equations.
• The quadratic formula is: x = (-b ± √(b² - 4ac)) / (2a).

Systems of Equations

• A system of equations comprises two or more equations sharing the same set of variables.
• Solving such systems entails identifying the variable values that satisfy all equations.
• Substitution, elimination, or graphing methods can solve systems of equations.

Inequalities

• An inequality uses inequality symbols (<, >, ≤, ≥, ≠) to compare two expressions.
• Inequalities define a range of acceptable values for a given variable.
• Solving inequalities mirrors solving equations, with the key exception of sign flipping upon multiplying/dividing by negative numbers.

Polynomials

• Polynomials include variables and coefficients, combined through addition, subtraction, multiplication, and non-negative integer exponents.
• The general form of a polynomial of degree n: aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀.
• Polynomials support addition, subtraction, multiplication, and division.

Factoring Polynomials

• Factoring simplifies a polynomial into a product of simpler polynomials.
• Common factoring techniques include identifying the greatest common factor (GCF), recognizing the difference of squares, and trinomial factoring.
• Factoring is useful to solve polynomial equations and simplify expressions.

Exponents and Radicals

• Exponents denote repeated multiplication of a base (e.g., x³ = x * x * x).
• Radicals, such as square roots, reverse the effect of exponents.
• Key exponent rules: xᵃ * xᵇ = xᵃ⁺ᵇ, (xᵃ)ᵇ = xᵃᵇ, xᵃ / xᵇ = xᵃ⁻ᵇ.

Rational Expressions

• A rational expression presents a fraction with polynomials in the numerator and denominator.
• Factoring and canceling common factors simplifies rational expressions.
• Numerical fractions share the same operational rules as rational expressions under addition, subtraction, multiplication, and division.

Functions

• Functions define a relationship between inputs and permissible outputs, where each input corresponds to exactly one output.
• The notation f(x) represents a function, with x as the input and f(x) as the output.
• The domain consists of all probable input values, while the range encompasses all potential output values.

Graphing

• Graphing uses a coordinate plane to visually represent algebraic relationships.
• Linear equations appear as straight lines on a graph.
• Quadratic equations are represented as parabolas when graphed.
• The x-intercept marks the point where the graph intersects the x-axis (y = 0).
• Conversely, the y-intercept is the point where the graph crosses the y-axis (x = 0).
• A line’s slope indicates its steepness and direction, calculated as "rise over run".

Logarithms

• Logarithms serve as the inverse operation of exponentiation.
• If bˣ = y, then log_b(y) = x.
• Logarithmic properties include: log_b(xy) = log_b(x) + log_b(y), log_b(x/y) = log_b(x) - log_b(y), log_b(x^p) = p*log_b(x).

Complex Numbers

• Complex numbers take the form a + bi, with a and b as real numbers, and i symbolizing the imaginary unit (i² = -1).
• Complex numbers are compatible with addition, subtraction, multiplication, and division.
• The conjugate of a + bi is expressed as a - bi.

Matrices

• Matrices form rectangular grids of numbers set out in rows and columns.
• Matrices can be added, subtracted, and multiplied, contingent on specific conditions.
• Matrices facilitate solving systems of linear equations and expressing linear transformations.
• Matrix algebra relies heavily on determinants and inverses.

Sequences and Series

• A sequence represents an ordered list of numbers.
• Arithmetic sequences show a consistent difference between successive terms.
• Geometric sequences demonstrate a uniform ratio between continuous terms.
• A series is the sum of the terms within a sequence.

Binomial Theorem

• The binomial theorem expands expressions like (a + b)ⁿ.
• Binomial coefficients, indicated as "n choose k" or ⁿCₖ, define expansion coefficients.
• Probability, statistics, and combinatorics benefit from using the binomial theorem.

Explore the basics of algebra, including variables, expressions, and equations. Learn how to solve equations using inverse operations and understand the concept of linear equations. This introduction covers essential algebraic principles.