In Exercises 1–4, is the algebraic expression a polynomial? If it is, write the polynomial in standard form. (2x+3)/x

Key Concepts

Polynomial Definition

A polynomial is an algebraic expression that consists of variables raised to non-negative integer powers and coefficients. It can be expressed in the form of a sum of terms, where each term is a product of a constant (coefficient) and a variable raised to a whole number exponent. For example, 3x^2 + 2x + 1 is a polynomial, while 1/x or x^-1 is not.

Standard Form of a Polynomial

The standard form of a polynomial is when the terms are arranged in descending order of their exponents. This means that the term with the highest degree appears first, followed by terms of lower degrees. For instance, the polynomial 4x^3 + 2x^2 - x + 5 is in standard form, as the terms are ordered from the highest degree (3) to the lowest (0).

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. In the expression (2x + 3)/x, the numerator is a polynomial, but the denominator is a variable, which means the entire expression is not a polynomial. Understanding the distinction between rational expressions and polynomials is crucial for determining whether an expression qualifies as a polynomial.