Introduction to Algebra: Core Concepts

• Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols.
• These symbols represent quantities without fixed values, known as variables.

Core Concepts in Algebra

• Variables are symbols (usually letters) representing unknown or changeable values.
• Constants are fixed values that do not change.
• Coefficients are numerical or constant quantities placed before and multiplying the variable in an algebraic expression.
• Operators are symbols (+, -, ×, ÷) indicating mathematical operations.
• Expressions are combinations of variables, constants, and operators, but do not contain an equals sign.
• Equations show the equality between two expressions, connected by an equals sign (=).
• Terms are parts of an expression or equation separated by operators.

Fundamental Operations

• Addition combines terms; only like terms (terms with the same variable raised to the same power) can be added directly.
• Subtraction finds the difference between terms, also requiring like terms for direct operation.
• Multiplication combines terms; coefficients and variables are multiplied separately.
• Division splits terms; coefficients and variables are divided separately.

Laws of Exponents

• Product of Powers: (a^m \cdot a^n = a^{m+n}) (when multiplying powers with the same base, add the exponents).
• Quotient of Powers: (a^m / a^n = a^{m-n}) (when dividing powers with the same base, subtract the exponents).
• Power of a Power: ((a^m)^n = a^{mn}) (when raising a power to another power, multiply the exponents).
• Power of a Product: ((ab)^n = a^n b^n) (the power of a product is the product of the powers).
• Power of a Quotient: ((a/b)^n = a^n / b^n) (the power of a quotient is the quotient of the powers).
• Zero Exponent: (a^0 = 1) (any non-zero number raised to the power of 0 is 1).
• Negative Exponent: (a^{-n} = 1/a^n) (a negative exponent indicates the reciprocal of the base raised to the positive exponent).

Polynomials

• Polynomials are expressions consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
• A monomial is a polynomial with one term.
• A binomial is a polynomial with two terms.
• A trinomial is a polynomial with three terms.
• Polynomials can be added, subtracted, multiplied, and divided, following algebraic rules.

Factoring

• Factoring is the process of breaking down a polynomial into the product of simpler polynomials or factors.
• Common Factoring: Identifying and extracting the greatest common factor (GCF) from all terms.
• 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: Grouping terms to find common factors and simplify.

Solving Equations

• Solving equations involves finding the value(s) of the variable(s) that make the equation true.
• Linear Equations are first-degree equations that can be written in the form (ax + b = 0).
• Quadratic Equations can be written in the form (ax^2 + bx + c = 0); these can be solved by factoring, completing the square, or using the quadratic formula (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}).
• Systems of Equations involve two or more equations with the same variables; solutions can be found by substitution, elimination, or matrix methods.

Inequalities

• Inequalities compare expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to).
• Solving inequalities is similar to solving equations, but multiplying or dividing by a negative number reverses the inequality sign.
• Compound Inequalities combine two or more inequalities, such as (a < x < b).

Functions

• A function is a relation between a set of inputs (domain) and a set of permissible outputs (range) with the property that each input is related to exactly one output.
• Linear Functions are functions with a constant rate of change and can be written in the form (f(x) = mx + b), where (m) is the slope and (b) is the y-intercept.
• Quadratic Functions are functions that can be written in the form (f(x) = ax^2 + bx + c), forming a parabola when graphed.

Radicals

• Radicals are expressions involving roots, such as square roots, cube roots, etc.
• Simplifying Radicals involves reducing the expression inside the radical to its simplest form.
• Rationalizing the Denominator is the process of removing radicals from the denominator of a fraction.
• Radical Equations are equations that contain radicals, which can be solved by isolating the radical and squaring (or cubing, etc.) both sides.

Logarithms

• Logarithms are the inverse operation to exponentiation, expressing the power to which a base must be raised to produce a given number.
• Common Logarithms use base 10, denoted as (log_{10}(x)) or simply (log(x)).
• Natural Logarithms use the base (e) (Euler's number, ≈2.71828), denoted as (log_e(x)) or (ln(x)).
• Logarithmic Properties include the product rule, quotient rule, and power rule, used to simplify logarithmic expressions and solve logarithmic equations.

Complex Numbers

• Complex Numbers are numbers that can be expressed in the form (a + bi), where (a) and (b) are real numbers, and (i) is the imaginary unit, defined as (i^2 = -1).
• Operations with Complex Numbers include addition, subtraction, multiplication, and division, following algebraic rules and the property (i^2 = -1).
• The Complex Conjugate of a complex number (a + bi) is (a - bi); multiplying a complex number by its conjugate results in a real number.

Explore the basics of algebra, including variables, constants, coefficients, and operators. Understand the difference between algebraic expressions and equations. Learn about fundamental operations such as addition and subtraction.