Monomial: Definition, Degree, Operations, And Examples

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Definition of Monomial

Monomial is one of algebra form that consists of one term. It is also defined as mathematical expression that consists of one term. The term is a number or the result of multiply between coefficient and the variables. Variable is symbol to replace unknown number yet. The general form is

General Form

axn

  • a is coefficient (real number and a ≠ 0)
  • x is variables (it can more than one variables)
  • n is degree or multiply of variables (not a fraction and n ≥ 0).

Difference between monomial and polynomial (binomial, trinomial, etc.) is polynomial consists of more than one term. For examples: x + 3, 2x3 + x2 – 1, etc.  

Based on the definition, monomial can consists of only one number or number with the variables. For example:

  • 3 (based on general form, it is 3x0 = 3.1 = 3)
  • 2x (based on general form, it is 2x1 = 2x
  • 4xy (based on general form, it is 4x1y1 = 4xy)
  • 5x2  (based on general form, it is 5x1x1 = 5.x.x = 5x2)
  • x2y (based on general form, it is 1x1x1y1 = 1.x.x.y = 1x2y = x2y)

Degree of a Monomial

Degree of monomial means the biggest power of the variable. It is same with degree in polynomial that means the biggest power of the variables. Every monomial can be written as the multiply of the coefficients and the variables.

Monomial forms are not only consists of one term but also have non-negative and non-fraction degree. If monomial form has more than one same variable, monomial degree is defined as the sum of the powers of it.

For example: 2x.x.x.y = 2x1+1+1y = 2x3y. 

Monomial Operation

Monomial operation consists of addition, subtraction, multiplying, and division.

  • Addition

Monomial can be added by the others if they have same variables and degree (same terms). For example:

  1. 2 and 3x can’t be added because of they have different in variable.
  2. x and 4x2 can’t be added because of they have different degree although they have same variable.
  3. 5x2 and x2 can be added because of they have same variable and degree. The result is 5x2 + x2 = 6x2.
  • Subtraction

Subtraction in monomial has same rules with the addition. Monomial can be subtracted if they have same variables and degree.

  • Multiplying

Monomial can be multiply without any rules although they have different variables and degrees.

  1. 2 and 3x can be multiplied and the result is 2 × 3x = 6x.
  2. x and 4x2 can be multiplied and the result is x × 4x2 = 4x1+2 = 4x3.
  3. 5x2 and x2 can be multiplied and the result is 5x2 × x2 = 5x2+2 = 5x4.
  • Division

Monomials can be divided without any rules although they have different variables and degrees. But remember that if they have negative and fraction degree, they aren’t monomial.

  1. 2 and 3x can be divided and the result is 2/3x = 2/3 x-1.
  2. 4x2 and x can be divided and the result is 4x2 ÷ x = 4x2-1 = 4x 
  3. 5x2 and x2 can be divided and the result is 5x2 ÷ x2 = 5.

Examples of Monomial’s Problems

1. Which one that is monomial form?

a. 12x or √x

b. 0,3 or x-3

c. x3/2 or 3/2 x

d. 2x – 1 or -2x

Answer

Monomial forms are:

a. 12x (because √x = x½ , monomial haven’t fraction degree)

b. 0,3 (because x-3 has negative degree, monomial degree ≥ 0).

c. 3/2 x (because 3/2 x has fraction degree)

d. -2x (because 2x – 1 is polynomial)

2. Find the result and define whether it monomial or not.

a. 2x – 3x = ….

b. -2x + xy = …

c. 2/3 x ÷ 2 = ….

d. y × y½ = ….

Answer

The results are:

a. 2x – 3x = -x (monomial)

b. -2x + xy = xy – 2x (can’t be operated because they have different variables. It is not monomial)

c. 2/3 x ÷ 2 = 2/3 × ½ = ⅓x (monomial)

d. y × y½ = y1+½ = y3/2 (it has fraction degree. It is not monomial)

Learn more

Cube Numbers and Cube Roots

Partial Sum

Geometric Sequences and Sums