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. $\left(2/3\right)t^6+\left(3/t^5\right)+1$

Verified step by step guidance

1. First, write down the given expression clearly: \(\frac{2}{3}t^{6} + \frac{3}{t^{5}} + 1\).
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. Terms with variables in the denominator or with negative exponents are not allowed in polynomials.
3. Examine each term: The first term \(\frac{2}{3}t^{6}\) is a constant multiplied by \(t^{6}\), which is valid in a polynomial. The second term \(\frac{3}{t^{5}}\) can be rewritten as \$3t^{-5}\(, which has a negative exponent, so this term is not allowed in a polynomial. The third term \)1$ is a constant term, which is allowed.
4. Since the expression contains a term with a negative exponent, the entire expression is not a polynomial.
5. Therefore, you do not need to determine the degree or classify it as monomial, binomial, or trinomial because the expression is not a polynomial.

Key Concepts

Definition of a Polynomial

A polynomial is an algebraic expression consisting of variables raised to non-negative integer exponents and coefficients, combined using addition, subtraction, and multiplication. Expressions with variables in denominators or with negative or fractional exponents 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 t^6 + 3t^2, the degree is 6.

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.