Algebraic Expressions

Algebraic expressions are the most commonly 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.

History of Algebraic Expressions
 

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. 

 

Components of Algebraic Expressions

 

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:

• 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.
  
   

•  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.

  
   

• 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.

  
   

• Terms: In an algebraic expression, we use terms which are connected using arithmetic operations, such as multiplication or division. 

  For example: \(6x^2 + 4x – 9\), where, 6x2, 4x, and -9 are different terms.
  
   

• 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:

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: 

• \(2 x\)
• \(8y^2\)
• \(9pq \)

Binomial

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

For example:

\(y + 4\)

\(4a – 7b\)

\(3p + 5q\)

Trinomial

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

For example: \(5a – 6b + 8c\) 

Multinomial

A multinomial is an expression containing two or more terms, which also includes binomials and trinomials.

For example:  \(5a^2 – 6ab + 8c + 4\)
 

Formulas in algebraic expressions help to simplify, expand, and factorize expressions easily. These identities and rules are very useful for solving equations and higher-level algebra problems.

Basic Algebraic Identities
 

• Square of a Sum: \((a+b)^2=a^2+2ab+b^2\)
   
• Square of a Difference: \((a−b)^2=a^2−2ab+b^2\)
   
• Product of a Sum and Difference: \((a+b)(a−b)=a^2−b^2\)
   
• Cube of a Sum: \((a+b)^3=a^3+3a^2b+3ab^2+b^3\)
   
• Cube of a Difference: \((a−b)^3=a^3−3a^2b+3ab^2−b^3\)
   
• Sum of Cubes: \(a^3+b^3=(a+b)(a^2−ab+b^2)\)
   
• Difference of Cubes: \(a^3−b^3=(a−b)(a^2+ab+b^2)\)

Expansion Formulas
 

• Square of a Trinomial: \((a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca)\)
   
• Cube of a Trinomial: \((a+b+c)^3=a^3+b^3+c^3+3(a+b)(b+c)(c+a)\)

Product Formulas
 

• Product of Two Binomials: \((x + a)(x + b) = x^2 + (a + b)x + ab\)
   
• General Product Rule: \((p + q)(r + s) = pr + ps + qr + qs\)

Factorization Formulas

• \(a^2+2ab+b^2=(a+b)^2\)
   
• \(a^2−2ab+b^2=(a−b)^2\)
   
• \(a^2 - b^2 = (a + b)(a - b)\)
   
• \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)
   
• \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)
   

Geometric Formulas
 

• Perimeter of square: \(4a\)
   
• Area of a square: \(a^2\)
   
• Area of a rectangle: \(l×b\)
   
• Area of a triangle: \(½  ×b×h\)
   
• Area of a circle: \(πr^2\)

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:

Addition of Algebraic Expressions

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:

\((2x + 5x) + (3y + 9y) + (6 + 8)\)

\(7x + 12y + 14\)

Subtraction of Algebraic Expressions

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: 

\((3a – 5a) + (4b + 6b) + (– 2 – 8)\)

\(–2a + 10b – 10\)

Multiplication of Algebraic Expressions

Terms can be expanded and multiplied by applying the distributive property 

\((9x + 4) (2x – 6)\)

Let’s multiply the terms using distributive property:

\((9x) (2x) + (9x) (– 6) + 4 (2x) + 4 (– 6)\)

\(18x^2 – 54x + 8x – 24\)

\(18x^2 – 46x – 24\).

Division of Algebraic Expressions

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.

• Monomial division: 
  
  \(\frac{16 y^3} {4y}\) \(= \frac{16}{4} = 4\)
  (Dividing coefficients)
  Now we subtract the exponents: \(y^3 ÷ y^1 = y^{3-1} = y^2\) \(= 4y^2\)
  
  
   
• Polynomial Division: 
   

  \((12x^2 + 6x) ÷  2x\)

  Now we will divide each term individually:

  \(\frac{12x^2}{2x} + \frac{6x}{2x}\)

   \(=6x + 3\)

Let us see why learning algebraic expressions are important for students. 

• Enhances logical and critical thinking: Learning algebraic expressions significantly improves students' logical thinking by systematically analyzing and simplifying them.
  
   
• Develops problem-solving skills: The concept teaches students to solve equations and problems in an organized, structured way, a foundational skill applicable across all subjects and real-life scenarios.
  
   
• Foundation for advanced mathematics: Algebraic expressions are the building blocks for understanding more complex mathematical concepts, including geometry, statistics, calculus, and higher math. 
  
   
• Preparation for standardized exams: Mastery of algebraic expressions is vital for success on math-based standardized tests and college entrance exams.
  
   
• Introduces the Concept of Variables: It helps children understand and work with unknown quantities (variables), a key step toward abstract mathematical reasoning.

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: \(5x + 6x – 8y + 7y = 11x + 15y\).
  
   
• Parents and teachers can utilize physical manipulatives or visual aids, like balance scales, algebra tiles, or drawing diagrams, to represent variables, constants, and equations. This makes abstract concepts more concrete.
  
   
• Teachers can promote a learning environment where students work in pairs or small groups to solve problems. This allows them to explain their thought process to peers, solidifying their own understanding.
  
   
• Parents and teachers can consistently help students to memorize the precise mathematical vocabulary (variable, coefficient, term, expression vs. equation). Make sure students can correctly identify the components before solving.
  
   
• Teachers, when grading or reviewing, give credit for correct steps and logical reasoning, even if the final answer is incorrect. This builds confidence in students and help to know the importance of the problem-solving process.
  
   
• Pay close attention to common mistakes, like incorrectly combining unlike terms or misapplying the distributive property, and provide targeted, immediate feedback to correct these fundamental errors.

Algebraic expressions are not just mathematical statements; they are extensively used in various fields.

Shopping and Discounts: Algebraic expressions can calculate the final price after discounts. For example, if an item costs ₹p and a discount of d% is applied, the price = \( p - \frac{dp}{100} \).

Mobile Recharge Packs: If a plan gives y GB per day, then in n days the total data is 𝑦 × 𝑛 = yn.

Exam Scores: If a test has q questions of 5 marks each, then the total marks is 5 × 𝑞 = 5q.

Work and Wages: If a worker earns ₹w per day, in d days the total wage is 𝑤 × 𝑑 = wd.

Fuel Consumption: If a car consumes f liters of fuel per kilometer, then for x kilometers it uses 𝑓 × 𝑥 = fx liters.