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. -5x¹¹

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First, recall the definition of a polynomial: it is an expression consisting of variables and coefficients, involving only non-negative integer exponents of the variables, combined using addition, subtraction, and multiplication.

Look at the given expression: \(-5x^{11}\). It has one term with a variable \(x\) raised to the power of 11, which is a non-negative integer.

Since there is only one term, this expression is a monomial (a polynomial with exactly one term).

The degree of a polynomial is the highest exponent of the variable in the expression. Here, the exponent is 11, so the degree is 11.

Therefore, \(-5x^{11}\) is a polynomial, specifically a monomial, and its degree is 11.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definition of a Polynomial

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

Degree of a Polynomial

The degree of a polynomial is the highest power (exponent) of the variable in the expression. For example, in -5x^11, the degree is 11 because the variable x is raised to the 11th power, which is the largest exponent present.

Classification by Number of Terms

Polynomials are classified based on the number of terms: 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 called a polynomial without these specific names.

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. -5x11