Math Lesson 11.1.1 - The Definition of Monomials

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Welcome to our Math lesson on The Definition of Monomials, this is the first lesson of our suite of math lessons covering the topic of The Definition of Monomials and Polynomials, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Definition of Monomials

In Chapter 6, we briefly mentioned the concept of monomers and polynomials. Thus, a monomial is an algebraic expression consisting of the product of a real number called a coefficient and one or more letters representing variables, which are raised to certain natural powers. For example,

are all monomials, as all variables are preceded by a number (coefficient), and moreover, the variables are raised to positive integer powers.

On the other hand,

are not monomials, as their variables are not always positive integers. Indeed, from the properties of indices and roots (more specifically, 1/xn = x - n and √x = x1/2), we can write the above expressions as

and

Since the index of x in the second expression and that of y in the first and third expressions are not positive integers (y has a negative integer index, while x has a rational index), they are not monomials.

Example 1

Which of the following algebraic expressions is a monomial?

1. 2/3 x⁴ yz⁵
2. 5x^1/3 yz³
3. -4/3 ab⁷
4. -3a√b/b^1/2

Solution 1

1. The expression
   2/3 x⁴ yz⁵
   is a monomial, because it contains a rational (and therefore real) number (2/3) as a coefficient, which precedes three variables: x, y and z, raised to the fourth, first and fifth power respectively (i.e. all variables are raised in natural powers).
2. The expression
   5x^1/3 yz³
   is not a monomial, because one of the variables, (variable x) is rational (therefore not natural), as its index is 1/3.
3. The expression
   -4/3 ab⁷
   is a monomial, because it contains a rational (and therefore real) number (-4/3) as a coefficient, which precedes two variables: a and b, raised to the first and seventh power respectively (i.e. all variables are raised in natural powers).
4. Apparently, the expression
   -(3a√b)/b^1/2
   is not a monomial as the indices of its variable b are not natural. However, by means of a few transformations (given the condition that b must not be zero), we obtain a monomial, because
   -(3a√b)/b^1/2 = -(3ab^1/2)/b^1/2 = -3a
   Hence, variable b is simplified.

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