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. $15a^2{b}^3+12a^3{b}^8-13b^5+12b^6$

Verified step by step guidance

1. Step 1: Understand the definition of a polynomial. A polynomial is an algebraic expression consisting of terms that are variables raised to non-negative integer exponents, multiplied by coefficients, and combined using addition or subtraction.
2. Step 2: Examine each term in the expression \$15a^{2}b^{3} + 12a^{3}b^{8} - 13b^{5} + 12b^{6}$. Check that all variables have whole number exponents and that there are no variables in denominators or under roots.
3. Step 3: Since all terms have variables with non-negative integer exponents and are combined by addition and subtraction, confirm that the entire expression is a polynomial.
4. Step 4: Count the number of terms in the polynomial. Here, there are four terms: \$15a^{2}b^{3}\(, \)12a^{3}b^{8}\(, \)-13b^{5}\(, and \)12b^{6}$. Since there are four terms, it is neither a monomial (1 term), binomial (2 terms), nor trinomial (3 terms).
5. Step 5: Determine the degree of the polynomial by finding the term with the highest sum of exponents. Calculate the degree of each term by adding the exponents of the variables in that term, then identify the largest sum as the degree of the polynomial.

Key Concepts

Definition of a Polynomial

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

Degree of a Polynomial

The degree of a polynomial is the highest sum of the exponents of the variables in any single term. For example, in the term 15a^2b^3, the degree is 2 + 3 = 5. The degree helps classify the polynomial and understand its behavior.

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.