In Exercises 1–4, is the algebraic expression a polynomial? If it is, write the polynomial in standard form. 2x+3x^2−5

Verified step by step guidance

1. Identify the given expression: \(2x + 3x^2 - 5\).
2. Check if each term in the expression is a polynomial term. A polynomial term is of the form \(ax^n\), where \(a\) is a constant and \(n\) is a non-negative integer.
3. The terms \(2x\), \(3x^2\), and \(-5\) are polynomial terms because they fit the form \(ax^n\) with non-negative integer exponents.
4. Since all terms are polynomial terms, the expression is a polynomial.
5. Write the polynomial in standard form by arranging the terms in descending order of their exponents: \(3x^2 + 2x - 5\).

Key Concepts

Polynomial Definition

A polynomial is an algebraic expression that consists of variables raised to non-negative integer powers and coefficients. It can include constants and can be expressed in the form of a sum of terms, where each term is a product of a coefficient and a variable raised to a power. For example, 2x^2 + 3x - 5 is a polynomial.

Standard Form of a Polynomial

The standard form of a polynomial is when the terms are arranged in descending order of their degrees, from the highest power to the lowest. For instance, the polynomial 3x^2 + 2x - 5 is in standard form because the term with the highest degree (x^2) is listed first, followed by the linear term (x) and the constant term.

Degree of a Polynomial

The degree of a polynomial is the highest exponent of the variable in the expression. It provides insight into the polynomial's behavior and the number of roots it may have. For example, in the polynomial 3x^2 + 2x - 5, the degree is 2, indicating that it is a quadratic polynomial.