1345 LearnersLast updated on June 10th, 2025
Algebraic Identities
The equations used in the algebraic expressions are the algebraic identities. The identities involve operations like addition, multiplication, subtraction, and division. It is used to solve the equations, analyze the data, and calculating area and volume.
What are Algebraic Identities in Math?
The LHS (left-hand side) and RHS (right-hand side) of algebraic identities are the same. The identities remain constant even after the value changes. Algebraic identities consist of variables, numbers, and operations. Some standard algebraic identities are
(a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 - 2ab + b2
a2 - b2 = (a + b)(a - b)
(x + a)(x + b) = x2 + (a + b)x + ab
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History of Algebraic Identities
The history of algebraic identities can be traced back to ancient civilizations like Babylonians and Egyptians. They used algebraic equations written in the form of words. Al-Khwarizmi, the Islamic mathematician, contributed to the development of algebraic equations. He is considered as the father of algebra.
In ancient Mesopotamia and Egypt civilizations, they solved the equations using methods like false position and a rudimentary understanding of algebraic concepts. Babylonian civilization is more advanced than the earlier civilizations. They tackled cubic and quadratic equations using the operations including addition and multiplication of the equations.
The algebra that we know now was developed by the Persian mathematician, Al-Khwarizmi. The modern notation and understanding of the binomial theorem were further developed during the Medieval and Renaissance. Rene Descartes and Pierre de Fermat are the modern mathematicians who later developed algebra by using modern notations and introducing alphabets as variables.
Properties of Algebraic Identities
The properties of algebra are the certain properties that help to solve the equations. The four basic properties of algebraic identities are
- Associative property
- Commutative property
- Distributive property
- Identity property
Associative property: The order of the variables in the group when adding or multiplying doesn't affect the answer.
Example;
In addition: (a + b) + c = a + (b + c)
In multiplication: (a × b) × c = a × (b × c)
Commutative property: The order of the variables doesn't change the result.
Example;
In addition: a + b = b + a
In multiplication: a × b = b × a
Distributive property: The product of multiplying a number with the sum of two or more numbers is the same as the sum of the product of multiplying the number with each addend.
Example: a × (b + c) = (a × b) + (b × a)
Identity property: Adding 0 to any number is the number itself. In multiplication if we multiply the number with 1 it results in the number itself.
Example:
In addition: a + 0 = a
In multiplication: a × 1 = a
Types of Algebraic Identities
Algebraic identities make it easier for students to solve algebraic equations. In this section let’s learn the types of identities.
Square of a Binomial
The binomial expression has only two terms in it. The square of a binomial is squaring a sum or difference of two terms. That is (a + b)2 = a2 + 2ab + b2 and (a - b)2 = a2 - 2ab + b2
Difference of Squares
The difference of squares is the difference between the perfect squares. That is a2 - b2 = (a + b) (a - b)
Cube of a Binomial
The cube of a binomial is the addition or subtraction of the third power of the binomial terms. The formulae are;
(a + b)3 = a3 + 3a2b + 3ab2 + b3
(a - b)3 = a3 - 3a2b + 3ab2 - b3
Sum of Cubes
The sum of cubes is the sum of two perfect cubes. In the form a3 + b3 = (a + b)(a2 - ab + b2)
Difference of Cubes
The difference between two perfect cubes. Which means a3 - b3 = (a - b)(a2 + ab + b2)
Perfect Square Trinomial
The result of squaring a binomial is of three terms which is a perfect square trinomial. That is a2 + 2ab + b2 = (a + b)2
Importance of Algebraic Identities for Students
To solve complex equations students use algebraic identities. So let’s learn the importance of algebraic identities for students.
- Algebraic identities are used to solve complex equations by breaking down the complex equations.
- For factorization of polynomial equations, students use algebraic identities
- Helps in understanding the logic behind algebraic identities and understand the basics of algebraic concepts.
Applications of Algebraic Identities
There are various applications of algebraic identities, like solving polynomial equations, simplifying expressions, expanding expressions, and factorization techniques. Let’s discuss them in detail.
Solving Polynomial Equations
Algebraic identities are used to solve the polynomial equation into simpler equations.
Examples, x2 - 9 = 0
Using the algebraic identities, a2 - b2 = (a - b)(a + b)
x2 - 9 can be written as x2 - 32
So, x2 - 32 = (x - 3)(x + 3) = 0
That is
(x - 3) = 0 or (x + 3) = 0
x = 3 or x = -3
Simplifying Expressions
Algebraic identities are used to simplify expressions, by reducing complexity and making it easier for the students.
For example, simplifying (x +3)2 - (x-3)2
Using square binomial that is (a + b)2 = a2 + 2ab + b2 and (a - b)2 = a2 - 2ab + b2
(x +3)2 = x2 + 6x + 9
(x - 3) = x2 - 6x + 9
That is (x2 + 6x + 9) - (x2 - 6x + 9)
x2 + 6x + 9 - x2 + 6x - 9 = 12x
Expanding Expressions
Expanding the expression using identities can make the calculation direct and easier.
For example, expanding (2 + 3)3
Using binomial expansion,
(a + b)3 = a3 + 3a2b + 3ab2 + b3
That is, (2 + 3)2 = 23 + 3 × (22) × 3 + 3 × 2 × 32 + 93
= 8 + 36 + 54 + 27 = 125.
Factorization Techniques
The equation is broken down into simpler equations to make them easier to solve.
For example, x3 - 8
Using the difference in cube a3 - b3 = (a - b)(a2 + ab + b2)
x3 - 8 can be written as x3 - 23
So, x3 - 23 = (x - 2)(x2 + 2x + 22)
Tips and Tricks to Master Algebraic Identities
By learning algebraic identities students can easily solve the complex equation easier. It is important for students to learn and understand their identities. To make the process easier let’s discuss some tips and tricks
Memorizing basic identities: By memorizing the basic algebraic identities students can easily understand the concept of algebraic identities. Such as (a + b)2 = a2 + 2ab + b2
(a - b)2 = a2 - 2ab + b2
a2 - b2 = (a + b)(a - b)
(x + a)(x + b) = x2 + (a + b)x + ab
Understanding the pattern: By understanding the pattern of the equations students can match it with known identities. For instance, x2 - 16 can be written as x2 - 42.
Real-World Applications of Algebraic Identities
Now we learn about algebraic identities so let's see how we use it in our real world. In the real world, we use it in the fields of engineering, physics, computer science, and so on.
Construction: Algebra is used in construction is used to calculate the elevation and design the building.
Budgeting: When calculating the exchange rates and interest rates we use algebra to make it easier.
Computer programming: Algebra is used to design algorithms, solve complex problems, and optimize codes.
Common Mistakes and How to Avoid Them in Algebraic Identities
When learning the algebraic identities to solve the expression students tend to repeat the same mistakes. In this section, let’s discuss some common mistakes and the ways to avoid them.
Mistake 1
Not adding the middle term in squaring a binomial
When expanding the binomial equation students forget to add the middle term. To avoid it students should always remember that the middle term is the product of two terms. That is (a + b)2 = a2 + 2ab + b2.
Mistake 2
Confusing with negative and positive cubes
While expanding the sum and negative cubes, students get confused. That is a3 + b3 = (a + b)(a2 + ab + b2) instead of a3 + b3 = (a + b)(a2 - ab + b2) . To avoid these errors students should memorize the identities. That is a3 + b3 = (a + b)(a2 - ab + b2) and a3 - b3 = (a - b)(a2 + ab + b2).
Mistake 3
Confusing with square and cube binomial
When expanding the squaring and cubing binomials students get confused with the expansion. That is expanding (a + b)3 = a3 + b3 instead of a3 + 3a2b + 3ab2 + b3, so students should memorize the standard identities to avoid these errors.
Mistake 4
Errors while expanding
When expanding an equation students tend to use wrong identities or make calculation errors. So they should double-check the answer and the identity they use.
Mistake 5
Confusing with factoring the perfect square trinomial
Confusing with the perfect square trinomial is common as they think any trinomial fits the form. The middle term of a perfect square trinomial is double the product of the two terms. That is, a2 + 2ab + b2 = (a + b)2.
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Solved Examples of Algebraic Identities
Problem 1
The length of a square garden is x + 4 units. Find the area of the square garden.
The area of the square is x2 + 8x + 16 square units
Explanation
The area of a square = s2
Here, s = x + 4 units
Therefore, area of the square = (x + 4)2
Using the identity, (a +b)2 = a2 + 2ab + b2
(x + 4)2 = x2 + 8x + 16
Thus, the area is x2 + 8x + 16 square units
Problem 2
The difference between the squares of two consecutive numbers is 15. Find the numbers?
The numbers are 7 and 8
Explanation
Considering the consecutive numbers as x and x + 1
The difference between the squares = (x +1)2 - x2 = 15
Using the identity, a2 - b2 = (a - b) (a + b)
So, (x +1)2 - x2 = (x + 1 - x) (x + 1 + x) = 1(2x + 1)
2x + 1 = 15
2x = 15 - 1
2x = 14
x = 7
Therefore, the numbers are 7 and 8.
Problem 3
Find the volume of a cube-water tank of side x + 2 unit
The volume of the water tank is x3 + 6x2 + 12x + 8 cubic unit
Explanation
The volume of the cube = side3
Here, side = (x + 2)3
Using the identity, (a +b)3 = a3 + 3a2b + 3ab2 + b3
(x + 2)3 = x3 + 3x2(2) + 3x(2)2 = 23
x3 + 6x2 + 12x + 8
The volume of the water tank is x3 + 6x2 + 12x + 8 cubic unit
Problem 4
Expand (a + b)3where a = 2x and b = 3
The expanded form is 8x3 + 36x2 + 54x + 27
Explanation
Using the cube of binomial identity (a +b)3 = a3 + 3a2b + 3ab2 + b3
That is (2x + 3)3 = (2x)3 + 3(2x)2(3) + 3(2x)(3)2 + 33
= 8x3 + 36x2 + 54x + 27
Problem 5
The sum of the cubes of two numbers is 855. If one number is 7, find the other number.
The other number is 8
Explanation
Given,
One number is 7
Let’s consider the other number as x
So, x3 + 73 = 855
x3 + 343 = 855
x3 = 855 - 343
x3 = 512
x = ∛512 = 8
So, the other number is 8
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FAQs on Algebraic Identities
1.What is the difference between an identity and an equation?
2.Can algebraic identities be used to simplify complex expressions?
3.Are there algebraic identities for cubes?
4.Why do we use algebraic identities in mathematics?
5.What is the square of a different identity?
6.How does learning Algebra help students in Philippines make better decisions in daily life?
7.How can cultural or local activities in Philippines support learning Algebra topics such as Algebraic Identities?
8.How do technology and digital tools in Philippines support learning Algebra and Algebraic Identities?
9.Does learning Algebra support future career opportunities for students in Philippines?
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Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
