Determine whether each expression is a polynomial. If the...

Determine whether each expression is a polynomial. If the polynomial has a specific name, for example, "monomial" or "binomial," give the name. If the expression is not a polynomial, explain why it is not.$\frac{1}{x-7}$

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Key Concepts

Polynomial Definition

A polynomial is an expression consisting of terms that are the product of constants and variables raised to nonnegative integer exponents. It cannot include operations like division by a variable because such operations could introduce negative or fractional exponents, which are not allowed in a polynomial.

Rational Expression

A rational expression is a ratio of two polynomials. When a variable appears in the denominator, it indicates that the expression might involve restrictions on its domain and it does not conform to the definition of a polynomial. In the given case, the presence of x in the denominator means that it is a rational expression, not a polynomial.

Negative Exponents

Expressions that result in negative exponents, often seen when variables are located in the denominator, are not considered polynomials. In a polynomial each term must have nonnegative integer exponents; any deviation from this criteria, such as the inherent negative exponent from rewriting a reciprocal, means the expression falls outside the polynomial classification.

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