Key Concepts in Mathematics

Arithmetic: Basic operations (addition, subtraction, multiplication, division).
Algebra: Symbols and letters to represent numbers and quantities in formulas and equations.
Geometry: Study of shapes, sizes, and properties of space.
Trigonometry: Relationships between angles and sides of triangles.
Calculus: Study of change and motion, involving derivatives and integrals.
Statistics: Collection, analysis, interpretation, and presentation of data.
Probability: Study of uncertainty and chance.

Pythagorean Theorem: In right triangles, a² + b² = c² (where c is the hypotenuse).
Fundamental Theorem of Algebra: Every polynomial equation has as many roots as its degree, counting multiplicities.
Mean Value Theorem: A function that is continuous on [a, b] and differentiable on (a, b) has at least one point c where the derivative equals the average rate of change over [a, b].

Order of Operations: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right) (PEMDAS).
Factoring: Breaking down an expression into simpler components (e.g., a² - b² = (a - b)(a + b)).
Functions: A relation where each input has exactly one output; commonly expressed as f(x).

Integer: Whole numbers (positive, negative, zero).
Rational Number: A number that can be expressed as a fraction a/b, where a and b are integers and b ≠ 0.
Irrational Number: A number that cannot be expressed as a simple fraction (e.g., π, √2).
Prime Number: A natural number greater than 1 that has no positive divisors other than 1 and itself.

Arithmetic: Involves fundamental operations such as addition, subtraction, multiplication, and division.
Algebra: Utilizes symbols to express relationships and formulate equations involving variables.
Geometry: Explores shapes, sizes, and spatial properties through various theorems and proofs.
Trigonometry: Analyzes relationships between angles and sides of triangles, important in various applications.
Calculus: Focuses on concepts of change and motion through derivatives (rate of change) and integrals (area under curves).
Statistics: Encompasses methods for collecting and interpreting data, essential for research and decision-making.
Probability: Examines the likelihood of events occurring, foundational for statistics and risk assessment.

Pythagorean Theorem: States that in right triangles, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a² + b² = c²).
Fundamental Theorem of Algebra: Asserts that every polynomial equation has the same number of roots as its degree, counting multiplicities.
Mean Value Theorem: Guarantees at least one point where the derivative of a continuous and differentiable function equals the average rate of change over an interval.

Order of Operations: Follows PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction) for evaluating expressions.
Factoring: Involves rewriting expressions into products of simpler factors, such as a² - b² = (a - b)(a + b).
Functions: Defined as a relationship between inputs and outputs, typically noted as f(x) for a specific output value for each input x.

Integer: Encompasses whole numbers, including negative numbers, zero, and positive numbers.
Rational Number: Can be expressed as a fraction a/b, where a and b are integers, and b is non-zero.
Irrational Number: Cannot be expressed as a simple fraction; examples include π and √2.
Prime Number: A natural number greater than 1 with no positive divisors other than 1 and itself.

Identify the Problem: Carefully read to understand what is required to solve.
Develop a Plan: Formulate a clear strategy or method for approaching the problem at hand.
Carry Out the Plan: Implement the steps of the selected strategy methodically.
Review/Reflect: Examine the solution for correctness and whether it makes sense contextually.

Linear Relationships: Can be described by the equation y = mx + b, where m is the slope and b is the y-intercept.
Quadratic Relationships: Expressed as y = ax² + bx + c, forms a parabolic graph.
Exponential Growth: Illustrated by functions of the form y = ab^x, representing rapid increase.

Science & Engineering: Essential for modeling phenomena, conducting analyses, and solving engineering challenges.
Finance: Pertains to interest calculations, investment analysis, and budgeting strategies.
Technology: Foundations of algorithms and data processing rely heavily on mathematical frameworks.

Engage in regular practice to enhance proficiency and problem-solving capabilities.
Use visual aids, such as graphs and diagrams, to clarify and reinforce understanding.
Work through example problems for practical application of concepts.
Collaborate with peers to share insights and learn different approaches to problem-solving.