Algebra Identities: Theories, Formulas

Algebra Identities: One of the most significant aspects of Elementary Mathematics is algebra. It begins in primary school and continues through senior secondary and even higher education. There are a thousand aspects when it comes to the importance of Algebraic Identities in Math. This article will provide definitions and examples to help you better comprehend their identities. Furthermore, these identities serve as the foundation for all algebra formulas.

Standard algebraic expressions and identities are equality conditions that apply to all of the variables’ values. In this article, we will discuss in simple terms the numerous algebraic identities of polynomials and trinomials. This will assist you in better understanding the various standard algebraic identities, which will help you improve your mathematical computation. Continue reading this article to know more about all algebraic identities, class 8 and all algebraic identities, class 9.

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All Algebraic Identities: Definition & Example of Algebra Identities

Let us consider a simple identity as below:

(a + b)² = a² + 2ab + b²

If an identity holds for every value of its variables, then we can easily substitute one side of equality with the other side. This means that if we found (a + b)² in other conditions, then we can replace it with a² + 2ab + b² and vice-versa. Therefore, we can use these shortcuts to easily manipulate algebra. Given below are the lists of all identities of algebra that are used commonly.

Algebraic Identities for Class 8: Algebra Identities Under Binomial Theorem

The Binomial Theorem hands out a standard way of expanding the powers of binomials or other terms. The general form of such algebra identities are mentioned below:

\((a+b)^n={}^{n}\textrm{C}_{0}\:a^n+{}^{n}\textrm{C}_{1}\:a^{n-1}.b+{}^{n}\textrm{C}_{2}\:a^{n-2}.b^2+…+{}^{n}\textrm{C}_{n-1}\:a.b^{n-1}+{}^{n}\textrm{C}_{n}\:b^n\)

1. Identity-I: (a + b)² = a² + 2ab + b²
2. Identity-II: (a – b)² = a² – 2ab + b²
3. Identity-III: (a + b)³ = a³ + 3a²b + 3ab² + b³
4. Identity-IV: (a – b)³ = a³ – 3a²b + 3ab² – b³
5. Identity-V: (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
6. Identity-VI: (x + a)(x + b) = x² + (a + b) x + ab
7. Identity-VII: (a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴
8. Identity-VIIII: (a – b)⁴ = a⁴ – 4a³b + 6a²b² – 4ab³ + b⁴

Now, let’s move on to the next algebraic identities.

Factoring Algebra Identities

These identities can be given as under:

1. Identity-I: a² – b² = (a + b) (a – b)
2. Identity-II: a³ – b³ = (a – b) (a² + ab + b²)
3. Identity-III: a³ + b³ = (a + b) (a² – ab + b²)
4. Identity-IV: a⁴ – b⁴ = (a² – b²) (a² + b²)

Trinomial Algebra Identities

The corresponding equalities are of trinomial algebra identities. You can derive such identities simply by factoring and manipulating the terms (given below):

1. Identity-I: (a + b) (a + c) (b + c) = (a + b + c) (ab + ac + bc) – abc
2. Identity-II: a² + b² + c² = (a + b + c)² – 2(ab + ac + bc)
3. Identity-III: a³ + b³ + c³ – 3abc = (a + b + c) (a² + b² + c² – ab – ac – bc)
4. Identity-IV: (a – b) (a – c) = a² – (b + c)a + bc

These identities will help you manipulate the algebraic equalities and will assist you in solving many types of mathematical expressions.

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Solved Examples on Identities of Algebraic Expressions

Let’s see some algebraic identities with examples.

Question 1: Find the product of (x + 2)(x + 2) using standard algebraic identities.

Solution: We can write (x + 2)(x + 2) as (x + 2)². We know that (a + b)² = a² + b² + 2ab.
So putting the value of a = x and b = 2, we get
(x + 2)² = x² + 2² + 2.2.x
= x² + 4 + 4x

Question 2: Factorize 25x² + 16y² + 9z² – 40xy + 24yz – 30zx using standard algebraic identities.

Solution: 25x² + 16y² + 9z² – 40xy + 24yz – 30zx is of the form: (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
Here, a = 5x, b = -4y, and c = -3z
Putting the values of a, b, and c in the equation, we get:
25x² + 16y² + 9z² – 40xy + 24yz – 30zx = (5x – 4y – 3z)²

Question 3: Expand (x – 3y)³ using standard algebraic identities.

Solution: (x– 3y)³ is of the form: (a – b)³ = a³ – 3a²b + 3ab² – b³
Here, a = x and b = 3y
Putting the values of a and b in the equation, we get:
(x– 3y)³ = x³ – 3.x².3y + 3.x.(3y)² – (3y)³
= x³ – 9x²y + 27xy² – 9y³

Question 4: Factorize 8x³ + 27y³ + 125z³ – 60xyz using standard algebraic identities.

Solution: 8x³ + 27y³ + 125z³ – 60xyz is of the form: a³ + b³ + c³ – 3abc = (a + b + c) (a² + b² + c² – ab – ac – bc)
Here, a = 2x, b = 3y, and c = 5z
Putting the values of a, b, and c in the equation, we get:
8x³ + 27y³ + 125z³ – 60xyz = (2x + 3y + 5z)[(2x)² + (3y)² + (5z)² – 2x.3y – 3y.5z – 2x.5z]
= (2x + 3y + 5z)(4x² + 9y² + 25z² – 6xy – 15yz – 10xz)

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List of Important Maths Formulas

You can make use of the important mathematical formula list prepared by Embibe to prepare well for your examinations. Apart from these algebra identities, you can find the other formulas as well in the table given below. These formulas can prove beneficial for your further exams.

Important FAQs

You can find the important FAQs related to these algebra identities:

Q1. What are the algebraic identities?
A. In simple words, an algebraic identity comprises any equation that comes true for any value given to its variable. You can make use of the examples related to such identities given in this article.

Q2. For which exams do these algebra identities hold importance?
A. Algebra identities are important for your K12 exams as well as other competitive examinations. The significance of these identities is such that even the top exams such as CAT, GATE, IAS, Banking, etc. ask questions from Algebra.

Q3. Who discovered algebraic identities?
A. The discovery of Algebraic Identities can be traced back to the medieval period. The Arabs and Central Asians were the people behind its discovery.

Q4. Where are algebraic identities used?
A. The algebra identities can be used in a lot of mathematical calculations. These can be related to factorization, trigonometry, integration and differentiation, quadratic equations, and more.

Q5. What is the best way to learn algebraic identities?
A. Practice is the key to master any mathematical concept. The best way to learn and master algebraic identities is to understand the formulas and implement them to solve questions. The more you practice, the better you get.

Now you know everything about Algebra Identities. We hope this detailed article helps you. Make use of Embibe’s study material, practice questions, and mock tests, which are available for, and ace your next exam.

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