In Exercises 1–4, is the algebraic expression a polynomial? If it is, write the polynomial in standard form. 2x+3x²−5

Verified step by step guidance

1. Step 1: Recall the definition of a polynomial. A polynomial is an algebraic expression consisting of terms that are sums or differences of variables raised to non-negative integer powers, multiplied by coefficients. For example, terms like 3x^2, -5x, and 7 are valid in a polynomial.
2. Step 2: Examine the given expression: 2x + 3x^2 - 5. Identify each term and check if it satisfies the criteria for a polynomial. The terms are 2x (a variable raised to the power of 1), 3x^2 (a variable raised to the power of 2), and -5 (a constant term). All terms meet the criteria for a polynomial.
3. Step 3: Write the polynomial in standard form. Standard form means arranging the terms in descending order of the powers of the variable. In this case, the term with the highest power is 3x^2, followed by 2x, and then the constant term -5.
4. Step 4: Rearrange the terms to write the polynomial in standard form: 3x^2 + 2x - 5.
5. Step 5: Verify that the polynomial is now in standard form by confirming that the terms are ordered by decreasing powers of the variable, and all coefficients and exponents are valid for a polynomial.

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 indicates the polynomial's behavior and the number of roots it can have. For example, in the polynomial 3x^2 + 2x - 5, the degree is 2, which means it is a quadratic polynomial and can have up to two real roots.