A polynomial is a mathematical expression consisting of variables, coefficients, and non-negative integer exponents combined using addition, subtraction, and multiplication. It has a finite number of terms. For example, [code]x² - 4x + 7[/code] is a polynomial with one variable, and [code]x³ + 2xyz² - yz + 1[br][/code] is a polynomial with three variables. Polynomials are fundamental in mathematics and are used in various applications, including solving equations, defining functions, and approximating other functions.[br][br]~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~[br][b][br]Key Concepts:[/b][br][br][b]Terms:[/b] A polynomial is made up of terms, which are separated by addition or subtraction. Each term consists of a coefficient (a number) and a variable part (a variable raised to a non-negative integer power). [br][br][b]Variables:[/b] These are symbols, usually letters, that represent unknown or changing values.[br][br][b]Coefficients:[/b] These are the numerical factors that multiply the variables in each term. [br][br][b]Exponents:[/b] These are the powers to which the variables are raised. In polynomials, exponents must be non-negative integers (0, 1, 2, 3, ...). [br][br][b]Degree:[/b] The highest power of the variable in a polynomial is called its degree. [br] [br]~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~[br][b][br]Types of Polynomials:[br][br][/b][b]Monomial:[/b] A polynomial with only one term (e.g., 5x², -3y, 7). [br][br][b]Binomial:[/b] A polynomial with two terms (e.g., x + 2, 3y² - 4z). [br][br][b]Trinomial:[/b] A polynomial with three terms (e.g., x² + 2x + 1, 2p² - 7p + 3). [br][br][b]~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~[br][br]Polynomials can also be classified by their degree:[/b] [br][br][b]Zero Polynomial:[/b] Degree 0 (e.g., 5). [br][br][b]Constant Polynomial:[/b] Degree 0 (same as zero polynomial) (e.g., -2). [br][br][b]Linear Polynomial:[/b] Degree 1 (e.g., 2x + 1). [br][br][b]Quadratic Polynomial:[/b] Degree 2 (e.g., x² - 3x + 2). [br][br][b]Cubic Polynomial:[/b] Degree 3 (e.g., x³ + 2x² - x + 5). [br][br]~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~[br][br][b]Examples:[br][br][/b]3x + 2 is a linear binomial, 5x² - 2x + 1 is a quadratic trinomial, 7 is a constant monomial, and x³ - 8 is a cubic binomial. [br][br]~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~[br][b][br]Polynomials are used in various mathematical and scientific fields like:[br][br][/b]Polynomial equations (equations where one side is a polynomial) are used to model and solve problems in many areas. [br][br]Polynomial functions are used to represent relationships between variables and are fundamental in calculus and other areas of mathematics.[br][br]Polynomials can be used to approximate more complex functions that are difficult to work with directly. [br][br]These are advanced mathematical concepts used in algebra and algebraic geometry. [br][br]~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~