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.See Example 1. -7z^5-2z^3+1

Verified step by step guidance

1. Step 1: Identify the expression: \(-7z^5 - 2z^3 + 1\).
2. Step 2: Check if the expression is a polynomial. A polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
3. Step 3: Confirm that \(-7z^5 - 2z^3 + 1\) is a polynomial because it involves only addition, subtraction, and non-negative integer exponents.
4. Step 4: Determine the degree of the polynomial. The degree is the highest power of the variable in the polynomial. Here, the highest power of \(z\) is 5, so the degree is 5.
5. Step 5: Classify the polynomial based on the number of terms. A monomial has 1 term, a binomial has 2 terms, and a trinomial has 3 terms. This expression has 3 terms, so it is a trinomial.

Key Concepts

Polynomial Definition

A polynomial is a mathematical expression consisting of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. Each term in a polynomial is formed by multiplying a coefficient (a constant) by a variable raised to a power. For example, -7z^5, -2z^3, and 1 are terms of a polynomial.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the expression. It indicates the polynomial's behavior and the number of roots it can have. For instance, in the polynomial -7z^5 - 2z^3 + 1, the degree is 5, as the term with the highest exponent is -7z^5.

Types of Polynomials

Polynomials can be classified based on the number of terms they contain. A monomial has one term, a binomial has two terms, and a trinomial has three terms. For example, -7z^5 is a monomial, while -7z^5 - 2z^3 + 1 is a trinomial, as it contains three distinct terms.