Algebra Identities: Algebra is one of the important components of Elementary Mathematics. It is introduced at the primary education level and goes on to senior secondary and even higher. If we talk about the importance of Algebraic Identities in Maths, then there can be a thousand points. This article will help you understand those identities a little better along with a few definitions and examples. Moreover, these identities form the basis for all the algebra formulas.
Standard algebraic expressions and identities are equality conditions that specifically hold for all the values of its variables. In this article, we will talk about the various algebraic identities of Polynomials and trinomials in simple terms. This will help you to understand the different standard algebraic identities which will further assist you in taking your mathematical computation to a stronger level.
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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 the 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 the algebra. Given below are the lists of all identities of algebra that are used commonly.
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:
- Identity-I: (a + b)2 = a2 + 2ab + b2
- Identity-II: (a – b)2 = a2 – 2ab + b2
- Identity-III: (a + b)3 = a3 + 3a2b + 3ab2 + b3
- Identity-IV: (a – b)3 = a3 – 3a2b + 3ab2 – b3
- Identity-V: (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
- Identity-VI: (x + a)(x + b) = x2 + (a + b) x + ab
- Identity-VII: (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4
- 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:
- Identity-I: a2 – b2 = (a + b) (a – b)
- Identity-II: a3 – b3 = (a – b) (a2 + ab + b2)
- Identity-III: a3 + b3 = (a + b) (a2 – ab + b2)
- 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):
- Identity-I: (a + b) (a + c) (b + c) = (a + b + c) (ab + ac + bc) – abc
- Identity-II: a2 + b2 + c2 = (a + b + c)2 – 2(ab + ac + bc)
- Identity-III: a3 + b3 + c3 – 3abc = (a + b + c) (a2 + b2 + c2 – ab – ac – bc)
- 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.
Algebra Identities: 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 free, and ace your next exam.
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