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. (2/3)t^6+(3/t^5)+1

Verified step by step guidance

1. Step 1: Identify the terms in the expression: \( \frac{2}{3}t^6 \), \( \frac{3}{t^5} \), and \( 1 \).
2. Step 2: Check if each term is a polynomial term. A polynomial term should have non-negative integer exponents.
3. Step 3: The term \( \frac{2}{3}t^6 \) is a polynomial term because it can be written as \( \frac{2}{3}t^6 \) with a non-negative integer exponent.
4. Step 4: The term \( \frac{3}{t^5} \) is not a polynomial term because it can be rewritten as \( 3t^{-5} \), which has a negative exponent.
5. Step 5: Since not all terms are polynomial terms, the entire expression is not 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. Expressions that contain negative exponents, fractional exponents, or variables in the denominator are not considered polynomials.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the expression. It provides insight into the polynomial's behavior and shape when graphed. For example, in the polynomial 2t^6 + 3t^5 + 1, the degree is 6, which indicates that the term with the highest exponent dominates the polynomial's behavior as the variable approaches infinity.

Classification 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. If a polynomial has more than three terms, it is simply referred to as a polynomial. This classification helps in understanding the structure and complexity of the polynomial.