Polynomials

A polynomial is a foundational concept in algebra and calculus, used to describe relationships between quantities.

Definition

A polynomial is an expression consisting of variables and coefficients, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

A polynomial of one variable (also called univariate polynomial) is generally written in the form:

a_n*x^n + a_{n-1}*x^{n-1} + ... + a_2*x^2 + a_1*x + a_0

Where:

'x' is the variable,
'n' is the degree of the polynomial,
'a_n', 'a_{n-1}', ..., 'a_2', 'a_1', 'a_0' are the coefficients.

For example, 5x^3 - 2x^2 + x - 7 is a polynomial of degree 3.

Properties

  1. Degree: The degree of a polynomial is the highest exponent of the variable in the polynomial. The degree indicates the number of solutions or roots the polynomial has.

  2. Leading Coefficient: The leading coefficient is the coefficient of the term with the highest degree.

  3. Roots or Zeros: The roots (or zeros) of a polynomial are the values of x that make the polynomial equal to zero.

  4. Factoring: Polynomials can often be factored into the product of lower-degree polynomials.

Applications

Polynomials are used extensively in mathematics and science, including in the fields of algebra, calculus, physics, and engineering. They are used to model behaviors, solve equations, and approximate functions.