 GeeksforGeeks App
Open App Browser
Continue

# Algebraic Expressions

Algebraic expressions are mathematical statements that are used to explain various conditions. We use variables and constants along with mathematical operators to define the algebraic expressions. Algebraic expressions are used to define unknown conditions in real life. Suppose we have to find the age of Arum if the age of his sister is twice the age of Arun and the sum of their age is 24 years. This situation can be easily explained using the Algebraic Expressions let the age of Arun be x then the age of his sister be 2x and the sum of their ages is x + 2x = 24 this is an algebraic expression,

## What are Algebraic Expressions?

Algebraic expressions are the expressions obtained from the combination of variables, constants, and mathematical operations like addition, subtraction, multiplication, division, and so on. An algebraic expression is made up of terms, there can be one or more than one term present in the expression.

### Examples of Algebraic Expressions

Some examples of algebraic expressions are,

• 5x + 4y
• 11x – 12
• 2y – 13, etc.

Here the above three expressions are the algebraic expression they have variables x, and y and constant terms -12, and -13.

## Constants, Variables, and Coefficients

In the algebraic expression, fixed numerals are called constants. Constants do not have any variables attached to them. For example, 3x – 1 has a constant -1 to it.

The image showing the constant and variables of an Algebraic Expression is added below, Variables are the unknown values that are present in the algebraic expression. For instance, 4y + 5z has y and z as variables.

Coefficients are the fixed values (real numbers) attached to the variables. They are multiplied by the variables. For example, in 5x2 + 3 the coefficient of x2 is 5.

A term can be a constant, a variable, or a combination of both. Each term is separated by either additioan or subtraction. For example, 3x + 5, 3x, and 5 are the two terms of the algebraic expression 3x + 5.

## Types of Algebraic Expressions

There are various types of algebraic expressions based on the number of terms in the algebraic expression. The three main types of algebraic expressions are,

• Monomial Expression
• Binomial Expression
• Polynomial Expression

### Monomial Expression

Monomial Expressions are algebraic expressions with only one term. For example, 2x, 5y, 3x2, 11xy, etc. all are monomial expressions.

### Binomial Expression

Binomial Expressions are algebraic expressions with two terms. For example, 2x + 3y, 5y + 11z, 3x2 + 4x, y +11xy, etc. all are binomial expressions.

2x + 3x is not a binomial expression as it can be further simplified as 5x.

### Polynomial Expression

Polynomial Expressions are algebraic expressions with more than two terms. For example, 2x + 3y + 11, 5y + 11z + 3x, 3x2 + 4x + 2, X + y +11xy, etc. all are polynomial expressions.

## Other Types of Algebraic Expressions

We can also categorize Algebraic Expressions into,

• Numeric Expression
• Variable Expression

Let’s learn about them in detail.

### Numeric Expression

An expression containing only numbers and not variables is called a numeric expression. They contain only numbers that are operated using various mathematical operators. For example 11 + 6, 15/2, 7 – 3, etc all are numeric expressions.

### Variable Expression

An expression containing both variables and numbers is called a variable expression. For example, 11x + 6y, 15x2 + 7, y – 3, etc all are numeric expressions.

The table below explains all the types of expression in brief.

## Simplifying Algebraic Expressions

Simplifying algebraic expressions is easy and very basic. First, understand what are like and unlike terms. Like terms have the same sign and unlike terms have opposite signs. To simplify the given algebraic expression, first, find out the terms having the same power. Then, if the terms are like terms, add them; if they are unlike terms, find the difference between the terms. The most simplified form of an algebraic expression is one where no same power terms are not repeated.

For instance, let’s simplify 4x5 + 3x3 – 8x2 + 67 – 4x2 + 6x3, the same powers that are repeated are cubic and square, upon combining them together, the expression becomes, 4x5 + (3x3 + 6x3) – (8x2 – 4x2) + 67. Now, simplifying the expression, the final answer obtained is 4x5 + 9x3 – 12x2 + 67. This term does not have any terms repeated that have the same power.

When an addition operation is performed on two algebraic expressions, like terms are added with like terms only, i.e., coefficients of the like terms are added.

Example: Add (25x + 34y + 14z) and (9x − 16y + 6z + 17).

Solution:

(25x + 34y + 14z) + (9x − 16y + 6z + 17)

By writing like terms together, we get

= (25x + 9x) + (34y − 16y) + (14z + 6z) + 17

By adding like terms, we get

= 34x + 18y + 20z + 17.

Hence, (25x + 34y + 14z) + (9x − 16y + 6z + 17) = 34x + 18y + 20z + 17.

### Subtraction of Algebraic Expressions

To subtract an algebraic expression from another, we have to add the additive inverse of the second expression to the first expression.

Example: Subtract (5b2 + 6b + 8) from (3b2 − 5b).

Solution:

(5b2 + 6b + 8) − (3b2 − 5b)

= (5b2 + 6b + 8) + (−3b2 + 5b)
= (5b2 − 3b2) + (6b + 5b) + 8
= 2b2 + 11b + 8

### Multiplication of Algebraic Expressions

When a multiplication operation is performed on two algebraic expressions, we have to multiply every term of the first expression with every term of the second expression and then combine all the products.

Example: Multiply (3x + 2y) with (4x + 6y − 8z)

Solution:

(3x + 2y)(4x + 6y − 8z)

= 3x(4x) + 3x(6y) − 3x(8z) + 2y(4x) + 2y(6y) − 2y(8z)
= 12x2 + 18xy − 24xz + 8xy + 12y2 − 16yz
= 12x2 + 12y2 + 26xy − 16yz − 24xz

### Division of Algebraic Expressions

When we have to divide an algebraic expression from another, we can factorize both the numerator and the denominator, then cancel all the possible terms, and simplify the rest, or we can use the long division method when we cannot factorize the algebraic expressions.

Example: Solve: (x2 + 5x + 6)/(x + 2)

Solution:

= (x2 + 5x + 6)/(x + 2)

After factorizing (x2 + 5x + 6) = (x + 2) (x + 3)

= [(x + 2) (x + 3)]/(x + 2)
= (x + 3)

## Algebraic Formulas

The general algebraic formulas we use for solving the algebraic expressions or algebraic equations are,

• (x + a) (x + b) = x2 + x(a + b) + ab
• (a + b)2 = a2 + 2ab + b2
• (a – b)2 = a2 – 2ab + b2
• (a + b)2 + (a – b)2 = 2 (a2 + b2)
• (a + b)2 – (a – b)2 = 4ab
• a2 – b2 = (a – b)(a + b)
• (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
• (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)
• a3 + b3 + c3 – 3abc = (a + b + c)(a2 + b2 + c2 – ab – bc – ca))

## Algebraic Expressions Examples

Example 1: Find out the constant from the following algebraic expressions,

• x3 + 4x2 – 6
• 9 + y5

Solution:

Constants are the terms that do not have any variable attached to them, therefore, in the first case, -6 is the constant, and in the second case, 9 is the constant.

Example 2: Find out the number of terms present in the following expressions,

• 4x2 + 7x – 8
• 5y7 – 12

Solution:

Terms are separated by each other either by addition or subtraction sign. Therefore, in the first case, there are 3 terms and in the second case, there are 2 terms.

Example 3: Simplify the algebraic term, z5 + z3 – y6 + 7z5 – 8y6 + 34 + 10z3

Solution:

In the expression, there are terms with the same power and same variable that are repeated, first bring them together,

(z5 + 7z5) + (z3 + 10z3) – (y6 – 8y6) + 34.

Now, simplify the expression,

8z5 + 11z3 – 9y6 + 34.

Example 4: Add (13x2 + 11), ( – 25x2 + 26x + 42) and (–33x – 29).

Solution:

Let F = (13x2 + 11) + ( – 25x2 + 26x + 42) + (–33x – 29)

⇒ F = 13x2 – 25x2 + 26x – 33x + 11 + 42 – 29

F = –12x2 – 7x + 24

Hence, (13x2 + 11) + ( – 25x2 + 26x + 42) + (–33x – 29) = –12x2 – 7x + 24.

Example 5: Solve (5x + 4y + 6z)2 + (3y – 7x)2.

Solution:

Given,

Let F = (5x + 4y + 6z)2 + (3y – 7x)2

From algebraic formulae, we have

(a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca

(a – b)2 = a2 – 2ab + b2

⇒ F = (5x)2 + (4y)2 + (6z)2 + 2(5x)(4y) + 2(4y)(6z) + 2(6z)(5x) + [(3y)2 – 2(3y)(7x) + (7x)2]

⇒ F= 25x2 + 16y2 + 36z2 + 40xy + 48yz + 60zx +9y2 – 42xy + 49x2

Now, combine the like terms.

= 74x2 + 25y2 + 36z2 – 2xy + 48yz + 60zx

Hence, (5x + 4y + 6z)2 + (3y – 7x)2 = 74x2 + 25y2 + 36z2 – 2xy + 48yz + 60zx.

## FAQs on Algebraic Expressions

### Q1: What are Algebraic Expressions?

Algebraic Expressions are mathematical expression which is used to represent various mathematical condition they use variables and constants to define various real-life condition.

### Q2: What are Types of Algebraic Expressions?

The three main types of algebraic expressions are

• Monomial: A monomial is an expression that has only one non-zero term. 2xy, 5y3, 7a, 2b, etc are some examples of monomials.
• Binomial: A binomial is an expression that has two non-zero terms. For example, 2a + 3 has two monomials 2a and 3 and hence it is a binomial.
• Polynomial: A polynomial is an expression that has more than two non-zero, unlike terms. For example, x-4y+ 8z has three monomials x, 4y, and z, hence it is a polynomial.

### Q3: Are all Algebraic Expressions Polynomials?

Not all algebraic expressions are polynomials but all polynomials are algebraic expressions i.e. there exist some algebraic expressions that are not polynomials. Example: πx + 1 is an algebraic expression but not a polynomial.

### Q4: How Algebraic Expressions are Derived?

Algebraic expression is an idea of representing numbers using vrious variables such as x, y, z, etc. without specifying their actual values. In simple terms, an algebraic expression is a mathematical statement where variables have been combined using fundamental arithmetic operations.

For example, Ram’s age is three times the age of Akash, and their total age is 48. Express it as an algebraic equation.

Let the age of Akash be x then age of Ram age is 3x, now the required algebric expression is 3x + x = 48