Why can't a monomial have a negative exponent?

Question:

Why can't a monomial have a negative exponent?

Definition of a Monomial:

In mathematics, a polynomial is a mathematical expression made up of a sum of terms that are products of constants, variables, and/or positive integer powers of those variables. A monomial can be defined as a polynomial that has exactly one term, or as a single term of a polynomial. Thus, a monomial is a single term that is a product of a constant, variables, and/or positive integer powers of variables.

Answer and Explanation:

A monomial cannot have a negative exponent, because of its definition. A monomial is a single term of a polynomial, and a polynomial is a mathematical expression that only contains terms that are products of a constant, variables, and/or positive integer powers of those variables. Therefore, a polynomial cannot contain terms that have a variable in the denominator. Otherwise it would be a rational expression, not a polynomial.

If a monomial had a negative exponent, then by the definition of negative exponents, it can be rewritten as a mathematical expression with a variable in the denominator.

  • {eq}x^{-n}=\frac{1}{x^{n}} {/eq}

We know that polynomials do not contain terms that have a variable in the denominator. Thus, if a monomial contained a negative exponent, it could not be a term in a polynomial. Rather it would be a rational expression.

For example, consider the following expression.

  • 2x2y-5

By the rule of negative exponents, we can rewrite this as follows:

  • 2x2y-5={eq}\frac{2x^{2}}{y^{5}} {/eq}

The expression {eq}\frac{2x^{2}}{y^{5}} {/eq} is a rational expression, so it cannot be a term in a polynomial, because it contains a variable in the denominator. Thus, it does not satisfy the definition of a monomial, and we see why a monomial can't have a negative exponent.


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