Identify each expression as a polynomial or not a polynomial. For each polynomial, give the degree and identify it as a monomial, binomial, trinomial, or none of these. -7z⁵-2z³+1

Verified step by step guidance

1. First, write down the given expression clearly: \(-7z^5 - 2z^3 + 1\).
2. Recall that a polynomial is an expression consisting of variables raised to non-negative integer exponents, combined using addition, subtraction, and multiplication by constants. Check if each term fits this definition.
3. Identify the terms in the expression: \(-7z^5\), \(-2z^3\), and \(1\). Each term has a variable raised to a whole number exponent or is a constant, so this is a polynomial.
4. Determine the degree of the polynomial by finding the term with the highest exponent. Here, the highest exponent is 5 from the term \(-7z^5\), so the degree is 5.
5. Count the number of terms to classify the polynomial: there are three terms, so it is a trinomial.

Key Concepts

Definition of a Polynomial

A polynomial is an algebraic expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication by non-negative integer exponents. Terms with variables raised to negative or fractional powers, or variables in denominators, are not polynomials.

Degree of a Polynomial

The degree of a polynomial is the highest exponent of the variable in the expression. It indicates the polynomial's order and helps classify its behavior and graph shape. For example, in -7z^5 - 2z^3 + 1, the degree is 5 because the highest power of z is 5.

Classification by Number of Terms

Polynomials are classified based on the number of terms: a monomial has one term, a binomial has two, and a trinomial has three. If there are more than three terms, it is simply called a polynomial without a special name. The given expression has three terms, so it is a trinomial.