Algebraic Expressions

What is an Algebraic Expression?

An algebraic expression in mathematics is an expression which is made up of variables and constants, along with algebraic operations (addition, subtraction, etc.). Expressions are made up of terms. They are also termed as algebraic equations.

Examples

3x + 4y – 7,  4x – 10, etc.

These expressions are represented with the help of unknown variables, constants and coefficient. The combination of these three (as terms) is said to be an expression. It is to be noted that, unlike the algebraic equation, an algebraic expression has no sides or equal to sign. Some of its examples include

• 3x + 2y – 5
• x – 20
• 2x− 3xy + 5

Variables, Coefficient & Constant

In Algebra we work with Variable, Symbols or Letters whose value is unknown to us. In the above expression (i.e. 5x – 3),

• x is a variable, whose value is unknown to us which can take any value.
• 5 is known as the coefficient of x, as it is a constant value used with the variable term and is well defined.
• 3 is the constant value term which has a definite value.

The whole expression is known to be the Binomial term, as it has two unlikely terms.

Types of Algebraic expression

There are 3 main types of algebraic expressions which include:

• Monomial Expression
• Binomial Expression
• Polynomial Expression

Monomial Expression

An algebraic expression which is having only one term is known as a monomial.

Examples of monomial expression include 3x4, 3xy, 3x, 8y, etc.

Binomial Expression

A binomial expression is an algebraic expression which is having two terms, which are unlike.

Examples of binomial include 5xy + 8, xyz + x3, etc.

Polynomial Expression

In general, an expression with more than one terms with non-negative integral exponents of a variable is known as a polynomial.

Examples of polynomial expression include ax by ca,  x3 + 2x + 3, etc.

Other Types of Expression:

Apart from monomial, binomial and polynomial types of expressions, an algebraic expression can also be classified into two additional types which are:

• Numeric Expression
• Variable Expression

Numeric Expression

A numeric expression consists of numbers and operations, but never include any variable. Some of the examples of numeric expressions are 10 + 5, 15 ÷ 2, etc.

Variable Expression

A variable expression is an expression which contains variables along with numbers and operation to define an expression. A few examples of a variable expression include 4x + y, 5ab + 33, etc.

Algebraic expression for Class 7

In Class 7, students will come across the terms of algebraic equations such as:

• Coefficient of a term
• Variables
• Constant
• Factors of a term
• Terms of equations
• Like and Unlike terms

Example of using these terms are given below.

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

Then, 2x2, 3xy, 4x and 7 are the terms

Coefficient of term x2 = 2

Constant term = 7

Example of like and unlike terms:

Like terms: 2x and 3x

Unlike terms: 2x and 3y

Factors of a term:

If 3xy is a term, then its factors are 3, x and y.

Monomial, Binomial & Trinomial

Also, in grade 7 we will learn about types of expressions, such as monomial, binomial and trinomial. Let us see examples of each.

Monomial: 2x

Binomial: 2x+3y

Trinomial: 2x+3y+9

Addition and Subtraction of Algebraic Expressions

We can add and subtract like terms easily.

Example: Add 3x + 5y – 6z and x – 4y + 2z.

By adding both the expressions we get;

(3x + 5y – 6z) + (x – 4y + 2z)

Separating the like terms and adding them together:

(3x + x) + (5y – 4y) + (-6z + 2z)

4x + y – 4z

Formulas

The general algebraic formulas we use to solve the expressions or equations are:

• (a + b)2 = a2 + 2ab + b2
• (a – b)2 = a2 – 2ab + b2
• a2 – b2 = (a – b)(a + b)
• (a + b)3 = a3 + b3 + 3ab(a + b)
• (a – b)3 = a3 – b3 – 3ab(a – b)
• a3 – b3 = (a – b)(a2 + ab + b2)
• a3 + b3 = (a + b)(a2 – ab + b2)

Solved Problem

Example: Simplify the given expressions by combining the like terms and write the type of Algebraic expression.(i) 3xy­­3 + 9x2 y3 + 5y3x

(ii) 7ab2 c2 + 2a3 b2 − 3abc – 5ab2 c2 – 2b2 a3 + 2ab

(iii) 50x3 – 20x + 8x + 21x3 – 3x + 15x – 41x3

Solution:

Creating a table to find the solution:

S.no Term Simplification Type of Expression
1 3xy­­3 + 9x2 y3 + 5y3x 8xy3 + 9x2y3 Binomial
2 7ab2 c2 + 2a3 b2 − 3abc – 5ab2 c2 – 2b2 a3 + 2ab 2ab2 c2 − 3abc + 2ab Trinomial
3 50x3 – 20x + 8x + 21x3 – 3x + 15x – 41x3 30x³ Monomial

How to derive algebraic expressions?

An algebraic expression is a combination of constants, variables and algebraic operations (+, -, ×, ÷). We can derive the algebraic expression for a given situation or condition by using these combinations.

For example, Sima age is thrice more than Tina. And the total age of Sima and Tina is 40. Expressing the algebraic form of this condition;

3x + x = 40 ⇒ 4x = 40; where x is the age of Tina.

Are algebraic expressions polynomials?

No, not all algebraic expressions are polynomials. But all polynomials are algebraic expressions. The difference is polynomials include only variables and coefficients with mathematical operations(+, -, ×) but algebraic expressions include irrational numbers in the powers as well.

Also, polynomials are continuous function (eg: x2 + 2x + 1) but algebraic expression may not be continuous sometimes (eg: 1/x2 – 1 is not continuous at 1).

Is 4 an algebraic expression?

No, 4 is not an algebraic expression because an expression should have at least one variable and one operation to be algebraic.

What are all the types of algebraic expressions?

There are three basic types of algebraic expressions. They are;

1. Monomial-Which has only one non-zero term.
2. Binomial-Which has two non-zero terms.
3. Polynomial- Which has more than one non-zero terms with non-negative integral exponents.

What is a variable in an algebraic expression?

A variable is an unknown quantity which we need to evaluate by solving the given algebraic equation.