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,

2x5; 1/2 x3 y; -4ab5; etc.

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,

3x/y2; 2√x; 2x3/y5; etc.

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

3x/y2 = 3 ∙ x1 ∙ y-2
2√x = 2 ∙ x1/2

and

2x3/y5 = 2 ∙ x3 ∙ y-5

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. 5x1/3 yz3
  3. -4/3 ab7
  4. -3a√b/b1/2

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
    5x1/3 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)/b1/2
    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)/b1/2 = -(3ab1/2)/b1/2 = -3a
    Hence, variable b is simplified.

More The Definition of Monomials and Polynomials Lessons and Learning Resources

Polynomials Learning Material
Tutorial IDMath Tutorial TitleTutorialVideo
Tutorial
Revision
Notes
Revision
Questions
11.1The Definition of Monomials and Polynomials
Lesson IDMath Lesson TitleLessonVideo
Lesson
11.1.1The Definition of Monomials
11.1.2The Definition of Polynomials
11.1.3The Degree of Polynomials
11.1.4The Names of Polynomials by Degree
11.1.5Finding the Value of Polynomials
11.1.6Finding the Zeroes of a Polynomial
11.1.7Finding the Zeroes of Polynomials through Iterative Methods
11.1.8The Homogenous and Non-Homogenous Polynomials

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