Polynomial

What is a Polynomial?

A polynomial is a mathematical expression consisting of one or more terms, each of which is the product of a constant (called the coefficient) and a non-negative integer power of one or more variables.

Structure

A polynomial comprises terms with a coefficient multiplied by a variable raised to an exponent, a non-negative integer. The coefficient is a numerical factor, while the variable can be something like x or y, and the exponent tells us how many times the variable is multiplied by itself.

This structure allows polynomials to represent a wide range of mathematical relationships in a compact form, enabling easy manipulation and calculation in various algebraic tasks.

No Negative Exponents

In a polynomial, the exponents of the variables must always be non-negative integers, meaning they can be 0, 1, 2, and so on, but never negative.

This rule ensures that the terms of a polynomial will not involve division by the variable, maintaining the polynomial’s structure of whole-number powers and keeping its behavior predictable and consistent across various mathematical operations.

Variable Powers

In mathematics, variables in a polynomial can be raised to different powers, reflecting how many times a variable is multiplied by itself. Each term of the polynomial may contain one or more variables, each potentially having its power.

Addition or Subtraction of Terms

Terms in a polynomial are added or subtracted, but there are no divisions by variables.

Examples of Polynomial

Constant Polynomial:
- $P(x)=5$ is a polynomial of degree $0$ (constant polynomial).
Linear Polynomial:
- $Q(x)=3x-2$ is a polynomial of degree $1$ (linear polynomial).
Quadratic Polynomial:
- $R(x)=2x^2+7x-1$ is a polynomial of degree $2$ (quadratic polynomial).
Cubic Polynomial:
- $S(x)=4x^3-x^2+3x+6$ is a polynomial of degree $3$ (cubic polynomial).

Not Polynomials

Negative Exponents:
- $T(x)=\frac{1}{x}$ is not a polynomial because it has a negative exponent.
Radical Expressions:
- $U(x)=\sqrt{2x+1}$ is not a polynomial because it involves a square root.
Divisions by Variables:
- $V(x)=\frac{2}{x}+3x$ is not a polynomial because it has a term with a variable in the denominator.