Monomial

The word “mono” means one, so expressions with a single term are called monomials. A monomial consists of a coefficient, one or more variables, and non-negative integer exponents. The degree of a monomial is the sum of the exponents of its variables. For example, in \( 8x^2y\), x and y are variables and 8 is the coefficient.

The expressions are classified into monomials, binomials, and trinomials based on the number of terms in them. The terms are parts of an expression separated by mathematical operations like addition and subtraction. Here, we will learn the difference between monomials, binomials, and trinomials. 
 

Monomials consist of one or more variables, their coefficients, and their degree. To better understand a monomial, it can be divided into the following: variable, coefficient, degree, and literal part. 

• Variable: Variables are letters used to represent unknown values in an expression. The value of expressions depends on the values of variables. 

• Coefficient: Numerical values that are multiplied by the variables in a term. For example, in the term 7x, 7 is the coefficient. 

• Degree: The degree of a monomial is the sum of the exponents of all variables. 

• Literal part: The literal part of a monomial is the part including the variables and their exponents. 

For example: 
 

Algebraic expressions with a single, non-zero term are called monomials. Now let's learn how to identify monomials. The monomials are identified using the following properties: 

• The monomials should have a single non-zero term.

• The exponents of monomials should be a whole number. 

• No variable should be in the denominator; that is, all variables should be in the numerator. 

• Examples for monomials: 5x²y, 6xy², etc.
  \(2xy + y\), \(5x^2 + 2x\), \(2x^{1/2} \), are not monomials. 

To factor a monomial, factor its coefficients and variables separately. When factoring monomials, we separate the coefficients and variables. Let’s learn how to factorize a monomial with an example. 

Example: Factor of the monomial 18x²y

Step 1: Identifying the coefficients and variables

Here, the coefficient is 18

Variables are x² and y.

Step 2: Factor the coefficient

Prime factorization of 18 = 2 × 3 × 3 .

Step 3: Factoring the variables

Factoring x² = x × x

Factoring y: y

The complete factorization of 18x²y is 2 × 3 × 3 × x × x × y

Basic operations like addition, subtraction, multiplication, and division can be performed on monomials. By following simple algebraic rules, we can perform operations on monomials.

Addition of Monomials

If the monomials have the same literal part, we can add them, and the result will be a monomial. We add the coefficients and then keep the literal part the same. 
For example: 5x²y + 15x²y = (5 + 15)x²y =20x²y.

Subtraction of Monomials

Like the addition of monomials, the subtraction of two monomials should have the same literal part. When subtracting two monomials, we first subtract the coefficients and then keep the literal part the same. 
For example: 20x² - 6x² = (20 - 6)x² = 14x^2.

Multiplication of Monomials

When multiplying monomials, multiply the coefficients and apply the law of exponents to the variables.

For example: 5x²y × 3xy = (5 × 3)(x² × x¹)(y¹ × y¹)

= 15x³y^2.

Division of Monomials 

Monomials with the same variables can be divided using the quotient law of exponents: (x^m / xⁿ = x^{m - n}). First, we divide the coefficients, and then we apply the quotient law of exponents to divide the variables. 

For example: 50x⁴y² / 5x²y
(50 / 5) (x⁴ / x²) (y² / y) = 10 × (x^{4 - 2}) × (y^{2 - 1}) 
= 10x²y 

Mastering monomials helps in simplifying algebraic expressions and solving equations efficiently. These tips make calculations faster and more accurate.

• A monomial is a single term consisting of a number, variable, or product of numbers and variables with whole number exponents.
   
• Always separate the coefficient (numerical part) from the variable(s) to simplify calculations.
   
• Apply rules like \(x^a \cdot x^b = x^{a+b} \quad \text{and} \quad (x^a)^b = x^{a \cdot b} \) to handle monomials efficiently.
   
•  Combine monomials with the same variable and exponent to simplify expressions.
   
• Multiply and divide monomials by separately handling coefficients and variables using exponent rules.

Monomials are the fundamental concept in algebra and are used in the fields such as physics, finance, biology, and so on. Here are some applications of monomials. 
 

• In finance, monomials are used in simple interest calculations. The simple interest is calculated using the formula, I = Prt. Here, prt is a monomial.
  
   
• The basic formula in mathematics and physics for calculating distance when time and speed are known is d = st. Here, the expression st is monomial.
  
   
• In biology and medicine, monomial expressions are used to prescribe dosages to patients. If the dosage is 10 mg per kg of the patient’s total body weight w, then the dosage is written as 10w, which is monomial.
  
   
• Monomials are used in calculating electrical power. For example, power \(P\) is calculated as \(P=VI\), where \(V\) is voltage and \(I\) is current. The expression \( VI\) is a monomial.
  
   
• Monomials are used to express concentrations in reactions. For instance, the amount of a substance can be written as \(C×V\), where \(C\) is concentration and \(V\) is volume. The expression \(CV\) is a monomial.