Algebra Basics Quiz

Definition: Branch of mathematics dealing with symbols and the rules for manipulating those symbols.
Key Concepts:
- Variables: Symbols that represent numbers. Commonly denoted as x, y, z.
- Expressions: Combinations of variables and constants (e.g., 2x + 3).
- Equations: Mathematical statements that two expressions are equal (e.g., 2x + 3 = 7).
- Functions: A relation between a set of inputs and a set of permissible outputs, usually represented as f(x).
Types of Equations:
- Linear Equations: Form ax + b = c; represents a straight line.
- Quadratic Equations: Form ax² + bx + c = 0; involves x² terms.
- Polynomial Equations: General form of an equation involving powers of x (e.g., ax^n + ... + k = 0).
Factoring: The process of breaking down an expression into products of simpler expressions.
Systems of Equations: Set of equations with the same variables; can be solved using substitution or elimination methods.

Definition: Branch of pure mathematics devoted to the study of the integers and more generally to objects built out of them.
Key Concepts:
- Prime Numbers: Natural numbers greater than 1 that have no positive divisors other than 1 and themselves.
- Composite Numbers: Natural numbers that have more than two distinct positive divisors.
- Greatest Common Divisor (GCD): The largest integer that divides two or more integers without leaving a remainder.
- Least Common Multiple (LCM): The smallest integer that is a multiple of two or more integers.
Divisibility Rules: Principles determining if one number is divisible by another (e.g., a number is divisible by 2 if it ends in 0, 2, 4, 6, or 8).
Congruences: A relation that indicates two numbers give the same remainder when divided by a specific number (modulus).
Fermat’s Last Theorem: States there are no three positive integers a, b, and c that satisfy the equation a^n + b^n = c^n for n > 2.
Applications: Number theory has applications in cryptography, computer science, and algorithms.

Branch of mathematics that manipulates symbols representing numbers and the rules governing them.
Variables: Symbols such as x, y, z that stand for unknown numbers.
Expressions: Combinations of variables and constants, exemplified by 2x + 3.
Equations: Statements asserting the equality of two expressions, e.g., 2x + 3 = 7.
Functions: Relationships between inputs and permissible outputs, commonly expressed as f(x).
Types of Equations:
- Linear Equations: Equations in the form ax + b = c, representing straight lines.
- Quadratic Equations: Equations such as ax² + bx + c = 0, involving squared terms.
- Polynomial Equations: General format involving powers of variables, represented as ax^n +...+ k = 0.
Factoring: Method to express an algebraic expression as a product of simpler expressions.
Systems of Equations: Collections of equations sharing the same variables, solvable through substitution or elimination.

Field of pure mathematics focusing on integers and their properties.
Prime Numbers: Natural numbers greater than one with only two divisors: 1 and the number itself.
Composite Numbers: Natural numbers with more than two distinct positive divisors.
Greatest Common Divisor (GCD): The largest integer that can divide two or more integers without leaving a remainder.
Least Common Multiple (LCM): The smallest multiple common to two or more integers.
Divisibility Rules: Guidelines to ascertain if one number is divisible by another; for instance, a number is divisible by 2 if it ends with 0, 2, 4, 6, or 8.
Congruences: Relations showing two numbers yield the same remainder when divided by a specific modulus.
Fermat’s Last Theorem: Proclaims no three positive integers can satisfy a^n + b^n = c^n for n greater than 2.
Applications: Integral in fields such as cryptography, computer science, and algorithm development.