Algebraic Expressions

Algebraic expressions is the most used way to solve unknown variables. Arithmetic operations (+, –, ×, and ÷) are performed on a combination of variables and constants to produce these mathematical expressions. For example: 2p + 3q + 5 (where p and q are variables; 5 is a constant and 2 and 3 are coefficients). An algebraic expression may contain one or more terms, and arithmetic operations help us in differentiating each term.
 

The origin of algebraic expressions is seen in ancient Babylonian mathematics. During that time, academicians applied symbols to indicate undefined quantities in equations. Diophantus, a Greek mathematician, brought in well-structured methods to resolve equations. Later, Al-Kwarizmi, a Persian mathematician known as the Father of Algebra, developed systematic solutions for quadratic and linear systems. It was the contributions of René Descartes that developed the algebraic expressions, which involved the use of the expressions in variables and symbols. In the 21st century, algebraic expressions are employed in many sectors such as engineering, computer application, and science. 
 

We now understand that algebraic expressions contain various mathematical symbols, numbers, or letters. Altogether, these elements are known as the components of algebraic expressions. Let’s discuss these components and how they define an expression’s structure:

i) Constants

Constants are the fixed numbers present in the expression. As the term suggests, their value does not change.
For example:
6y + 5 is an expression where 5 is a constant.

ii) Variables

The undefined symbols are represented using letters known as variables. The value of variables keeps changing in different problems.
For example: 
4x + 2 is an expression where x represents variable.

iii) Coefficients

There are numbers in a term that have variables in it. Such numbers are coefficients.
They show the number of times the variable in the term is multiplied.
For example: 
In 3p + 9q, the numbers 3 and 9 are coefficients of p and q.

iv) Terms

In an algebraic expression we use terms which are connected using arithmetic operations such as multiplication or division. 
For example: 6x2 + 4x– 9, where, 6x2, 4x, and -9 are different terms.

v) Operators

The arithmetic operations (+, –, ×, and ÷)  using which we connect the terms are referred to as operators. 
For example:
In the expression, 5a – 3b + 8, the operators are (–) and (+).
 

Algebraic expressions may vary based on the number of terms they have. The different types of algebraic expressions are given below:

i) Monomial

If the algebraic expression has only one term in it, the expression is Monomial. A monomial can be a variable, a constant, or a combination of a constant and a variable.

For example: 
2 x,
8y2
9pq 

ii) Binomial

The algebraic expression that holds 2 terms is known as a binomial. The terms are differentiated by subtraction or addition signs.

For example:
y + 4
4a – 7b
3p + 5q

iii) Trinomial

If the algebraic expression contains three terms, then it is called a Trinomial.

For example: 5a – 6b + 8c 

iv) Multinomial

A multinomial is an expression containing two or more terms, which also includes binomials and trinomials.
For example:  5a2 – 6ab + 8c + 4
 

Algebraic expressions are important for students as the concept improves their logical thinking skills. The concept of algebraic expressions help children understand complex concepts such as geometry and statistics. Learning algebraic expressions can help students in the preparation of math-based exams. They will also learn how to solve equations or problems in an organized way, developing problem-solving skills. 
 

We solve algebraic expressions using arithmetic operations. These mathematical operations help us in differentiating and simplifying the terms in an expression. We will now discuss the four operations of algebraic expression:

i) Addition

We use addition to add up the terms that have the same exponents and variables (like terms). 
For example: (2x + 3y + 6) + (5x + 9 y + 8).
To find the sum, we will combine the like terms:
(2x + 5x) + (3y + 9y) + (6 + 8)
  7x + 12y + 14

ii) Subtraction

In the subtraction method, we will first remove the negative sign and then combine the similar terms.
For example: (3a + 4b – 2) – (5a – 6b + 8)
Let’s first give out the negative sign:
3a + 4b – 2 –5a + 6b – 8
To subtract the expression, we combine the like terms: 
(3a – 5a) + (4b + 6b) + (– 2 – 8)
  –2a + 10b – 10

iii) Multiplication

Terms can be expanded and multiplied by applying the distributive property 
(9x + 4) (2x – 6)
Let’s multiply the terms using distributive property:
(9x. 2x) +(9x) (– 6) + 4 (2x) + 4(– 6)
18x2 – 54x + 8x – 24
  18x2 – 46x – 24.

iv) Division

The Division of algebraic expressions involves taking out and eliminating common terms. The division of algebraic expressions is in two ways : monomial and polynomial division.
 

16 y3/ 4y
=16/ 4 = 4 (Dividing coefficients)
Now we subtract the exponents: y3 ÷ y1 = y3-1 = y2
  4y2
 

(12x2 + 6x) ÷  2x
Now we will divide each term individually:
12x2/ 2x + 6x/ 2x
 6x + 3
 

Students find it challenging to solve the algebraic expressions. It can be due to several reasons. Here, we will discuss a few tips and tricks that may help students excel in algebraic expressions:

• Understanding algebraic expressions requires a strong foundation in its components such as variables, terms, operators, constants, and coefficients.
  
   
• Remember the acronym BODMAS (Brackets, Orders (exponents), Division, Multiplication, Addition, and Subtraction) while solving algebraic expressions.
  
   
• Learning the identities by memorizing them helps in grasping them quickly.
  
   
• Don’t skip any step while solving the expressions, as it may break the flow of the process.
  
   
• We can simplify the expressions by combining the like terms. For example: 5x + 6x – 8y +7y = 11x +15y.
   

Algebraic expressions are not just mathematical statements; they are extensively used in various fields. Algebra helps children in estimating the interest rates of their study loans or savings growth. It helps them understand the different load capacities by determining the unknown values. It helps them analyze their favorite player’s statistics and calculate the scores. If you are traveling, you can check the distance covered or the speed. It can be used in determining the area of land or the water needs.