How to derive an algebraic expression?

The basic concept of algebra taught us how to express an unknown value using letters such as x, y, z, etc. These letters are termed here as variables. this expression can be a combination of both variables and constants. Any value that is placed before and multiplied by a variable is termed a coefficient. An idea of expressing numbers using letters or alphabets without specifying their actual values is termed an algebraic expression.

Algebraic Expression

In mathematics, It is an expression that is made up of variables and constants along with algebraic operations such as addition, subtraction, etc. These Expressions are made up of terms. Algebraic expressions are the equations when the operations such as addition, subtraction, multiplication, division, etc. are operated upon any variable.

A combination of terms by the operations such as addition, subtraction, multiplication, division, etc is termed as an algebraic expression (or) a variable expression. Examples: 2x + 4y – 7, 3x – 10, etc.

The above expressions are represented with the help of unknown variables, constants, and coefficients. The combination of these three terms is termed as an expression. unlike the algebraic equation, It has no sides or ‘equals to’ sign.

Types of Algebraic expression

• Monomial Expression: An expression that has only one term is termed a Monomial expression. Examples of monomial expressions include 4x⁴, 2xy, 2x, 8y, etc.
• Binomial Expression: An algebraic expression which is having two terms and unlike are termed as a binomial expression. Examples of binomial include 4xy + 8, xyz + x², etc.
• Polynomial Expression: An expression that has more than one term with non-negative integral exponents of a variable is termed a polynomial expression. Examples of polynomial expression include ax + by + ca, x³ + 5x + 3, etc.

Some Other Types of Expression

Other expressions also present apart from monomial, binomial, and polynomial types of expressions which are,

• Numeric Expression: An expression that consists of only numbers and operations, but never includes any variable is termed a numeric expression. Some of the examples of numeric expressions are 11 + 5, 14 ÷ 2, etc.
• Variable Expression: An expression that contains variables along with numbers and operations to define an expression is termed a variable expression. Some examples of a variable expression include 5x + y, 4ab + 33, etc.

Some algebraic formulae

1. (a + b)² = a² + 2ab + b²
2. (a – b)² = a² – 2ab + b²
3. (a + b)(a – b) = a² – b²
4. (x + a)(x + b) = x² + x(a + b) + ab
5. (a + b)³ = a³ + b³ + 3ab(a + b)
6. (a – b)³ = a³ – b³ – 3ab(a – b)
7. a³ – b³ = (a – b)(a² + ab + b²)
8. a³ + b³ = (a + b)(a² – ab + b²)

There are some terms of algebraic expression which are basically used. Examples of using these terms,

• If 2x² + 3xy + 4x + 7 is an algebraic expression.

Terms: 2x², 3xy, 4x, and 7

Coefficient of term: 2 is the coefficient of x²

Constant term: 7

Variables: Here x, y are variables

Factors of a term: If 2xy is a term, then its factors are 2, x, and y.

Like and Unlike terms : Example of like and unlike terms:

• Like terms: 4x and 3x
• Unlike terms: 2x and 4y

How to derive an algebraic expression?

Answer:

An algebraic expression is derived from variables and constants using different operations.

It is an expression that is made up of variables and constants along with algebraic operations such as addition, subtraction, etc.. these Expressions are made up of terms. Algebraic expressions are the equations when the operations such as addition, subtraction, multiplication, division, etc. are operated upon any variable.

A combination of terms by the operations such as addition, subtraction, multiplication, division, etc is termed as an algebraic expression (or) a variable expression.

Examples: 2x + 4y – 7, 3x – 10, 4x + 7, etc.

Here, 4x + 7 is a term

x is a variable whose value is unknown and which can take any value.

Here, 4 is known as the coefficient of x, as it is a constant value used with the variable term.

7 is the constant value term that has a definite value.

Some formulas to derive algebraic expression,

(a + b)² = a² + 2ab + b²

(a – b)² = a² – 2ab + b²

And so on given above.

Sample Problems

Question 1: Solve for x = 4: x² – 4x + 5

Solution:

x² – 4x + 5

Here,

4² – (4 × 4) + 5

= 16 – 16 + 5

= 0 + 5

= 5

Question 2: simplify (4a + 2 )² + 54a

Solution:

= (4a + 2 )² + 54a

= {16a² + 4 + 2(4a)(2)} + 54a, {(a + b)² = a² + 2ab + b²}

= 16a² + 4 + 16a + 54a 

= 16a² + 70a + 4

Question 3: Identify the various components of algebraic expressions: 7x + 3y – 2.  

Solution:  

The algebraic expression is constituted by the following parts:  

7x and 3y, where 7 and 3 are coefficients and x and y are variables. -2 is the constant part of the expression.