In algebra, a monomial is a mathematical expression that consists of only one term. It can be represented as a combination of a constant, a variable, or a product of constants and variables raised to non-negative integer exponents.

The general form of a monomial is \(ax^n \), where:

  • \( a \) is a real number representing the coefficient. The coefficient can be positive, negative, or zero.
  • \( x \) is the variable.
  • \( n \) is a non-negative integer representing the exponent. It determines the degree of the monomial and if \( n = 0 \), the monomial reduces to a constant.

Regarding the properties of exponents, it is worth noting that if any variable is raised to the power of zero, the resultant value will always be \(1\). This particular rule also applies to monomials. Whenever any monomial is raised to the power of zero, the output will be \(1\). \[3x^0 = 1 \]

Examples of monomials include:

  • \( 4 \) a real number.
  • \( -2x \) a product of constant and variable.
  • \( 3x^2 \) a product of constants and variables raised to non-negative integer exponents.

Sum and difference

In polynomial algebra, the addition or subtraction of monomials is subject to certain constraints. Specifically, the variables and their corresponding exponents of the monomials under consideration must be identical for them to be combined. In such cases, the coefficients of the monomials can be aggregated through addition or subtraction while the variables and their exponents remain unchanged. For example:

  • \(3x + 2x = 5x\)
  • \(3x^2 + 2x \), the monomials in this expression cannot be directly added since they have different degrees.

Product

Monomials can be multiplied by multiplying their coefficients and adding the exponents of the like variables. This rule allows us to simplify expressions and easily solve equations involving monomials. For instance, the product of \( (3x^2)(2x^3) \) can be computed as follows: first, we multiply the coefficients, which gives us \( 3 \times 2 = 6\). Next, we add the exponents of the like variables, which in this case is \( 2+3 = 5 \). Thus, we get:

\[(2x^2)(3x^3) = 6x^{2+3} = 6x^5 \]


Division

When dealing with monomials, it is possible to divide them as long as their variables have the same base and the divisor’s exponent is less than or equal to the dividend’s exponent. As an example, consider the expression:

\[ \frac{6x^5}{2x^2} = 3x^{5-2} = 3x^3 \]

When dividing monomials with the same bases, the division cannot be performed directly if the divisor’s exponent is greater than the dividend’s exponent. In such cases, it is necessary to simplify the expression by subtracting the exponents and, if possible, further simplifying the resultant expression. For example

\[ \frac{x^2}{3x^3} = \frac{x}{3x^{(3-2)}} = \frac{1}{3x} \]


Powers

When a monomial is raised to a power, each term within the monomial is also raised to that power. For example, if we take the expression \((3x^3)^2\), we can apply this rule to get:

\[(3x^3)^2 = 3^2 \times x^{3 \times 2} = 9x^6\]