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Defining monomials

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As the prefix "mono-" implies, a monomial is a number multiplied by a variable (or more than one variable), each of whose variables are raised to a power. The non-variable component of a monomial is called a coefficient. Keep in mind that there are no negative monomials. The powers of all monomials must be nonnegative integers. This means that 0 can be used as part of a monomial, but negative numbers cannot. Therefore, it's not just that the powers monomials must be positive, but they must at least be nonnegative. Thus, zero is permissible. 7x^-4 is not a monomial because it is raised by a negative power. 5x^3 is a monomial because it is a number multiplied by a variable, whose variable is raised by a power, and that power is a nonnegative integer.

What is important to keep in mind is that the above rule only holds for the powers of the variables. The coefficient can itself be raised by a power. 2^-2x^4 is not problematic for monomials. Indeed, it is a monomial. The coefficient is itself raised by a negative power, but as long as none of the coefficient's variables are raised by a negative power, its status as a monomial is not compromised.

Things can sometimes get tricky. 4/x is not a monomial. But why? Because it is equivalent to 4x^-1. Keep in mind that expressions whose variables are raised to a negative power are not monomials. Therefore, this is not a monomial.

Keeping in mind what we mentioned before about the prefix "mono-" in monomial meaning "1," an expression such as 4x + x^8 is not a monomial because neither term can be combined with the other. But why does this disqualify it from being a monomial? To put it more positively, why does an expression whose two terms (each of which has variables raised to powers) are combinable count as a monomial? Because if they can be combined, that means they can be reduced or simplified to one term. Therefore, such an expression is a monomial.

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