The algebraic equations which are valid for all values of variables in them are called algebraic identities. They are also used for the factorization of polynomials. In this way, algebraic identities are used in the computation of algebraic expressions and in solving different polynomials. You have already learned about a few of them in the junior grades.
In this article, Eduriefy will recall them and introduce you to some more standard algebraic identities, along with examples.
Algebraic equations form the basis of all simple and complex formulas used in Mathematics. The algebraic identities are the equations in which all the values of variables are valid and the equation will hold true irrespective of any value of the variables. Algebraic identities find application in advanced mathematical equations, analysis, and research-based concepts. It is important for mathematics students to be proficient with these equations as they are very helpful in solving engineering and scientific problems.
Four standard algebraic identities are listed below:
Identity-1: Algebraic Identity of Square of Sum of Two Terms
(a+b)2=a2+2ab+b2
Identity-2: Algebraic Identity of Square of Difference of Two Terms
(a–b)2=a2–2ab+b2
Identity-3: Algebraic Identity of Difference of Two Squares
(a+b)(a–b)=a2–b2
Identity-4: Algebraic Identity (x+a)(x+b)
(x+a)(x+b)=x2+(a+b)x+ab
Algebraic Identities Under Binomial Theorem
The Binomial Theorem hands out a standard way of expanding the powers of binomials or other terms. The binomial theorem is used in algebra, probability, etc. The general form of such algebraic identities is mentioned below:
(a+b)2=nC0an+nC1an–1.b+nC2an–2.b2+….+nCn–1a.bn–1+nCnbn
PRACTICE EXAM QUESTIONS
1.) Using the area of a rectangle (length×breadth) and the area of the square (side)2 we can visualize the identity as follows:
The total area of the rectangle is the sum of areas of rectangles and squares in it.
⇒(x+a)(x+b)=x2+ax+bx+ab
⇒(x+a)(x+b)=x2+(a+b)x+ab
Proof of a2–b2=(a+b)(a–b)
Let us consider a square with side a=(a–b)+b units as shown in the figure.
The big square is divided into four quadrilaterals (rectangles, squares), as shown in the figure.
2.) Using the area of a rectangle (length×breadth) and the area of the square (side)2, we can visualize the identity as follows:
The total area of the square is the sum of areas of rectangles and squares in it.
⇒a2=a(a–b)+b(a–b)+b2
⇒a2=(a+b)(a–b)+b2
⇒a2–b2=(a+b)(a–b)
Proof of (a+b+c)2=a2+b2+c2+2ab+2bc+2ca
Let us consider a square with side a+b+c units, as shown in the figure.
The big square is divided into nine quadrilaterals (rectangles, squares), as shown in the figure.
3.) Using the area of a rectangle (length×breadth) and the area of the square (side)2, we can visualize the identity as follows:
The total area of the square is the sum of areas of rectangles and squares in it.
⇒(a+b+c)2=a2+ab+bc+ca+ba+cb+ac+c2+b2
⇒(a+b+c)2=a2+b2+c2+2ab+2bc+2ca
Algebraic Identities Chart
The chart of algebraic identities helps us to understand various types of identities, uses, and applications in algebra and other branches of mathematics. The chart includes:
- Square of Binomial
- Difference Between Squares
- Cube of Binomials
- Sum of Cubes
- Difference Between Cubes
- Product of Binomials
- Square of Trinomials
Standard Algebraic Identities List
All the standard Algebraic Identities are derived from the Binomial Theorem, which is given as:
Some Standard Algebraic Identities list are given below:
Identity I: (a + b)2 = a2 + 2ab + b2
Identity II: (a – b)2 = a2 – 2ab + b2
Identity III: a2 – b2= (a + b)(a – b)
Identity IV: (x + a)(x + b) = x2 + (a + b) x + ab
Identity V: (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
Identity VI: (a + b)3 = a3 + b3 + 3ab (a + b)
Identity VII: (a – b)3 = a3 – b3 – 3ab (a – b)
Identity VIII: a3 + b3 + c3 – 3abc = (a + b + c)(a2 + b2 + c2 – ab – bc – ca)
Solved Examples of Algebraic Identities
Example 1:
Find the product of (x + 1)(x + 1) using standard algebraic identities.
Solution:
(x + 1)(x + 1) can be written as (x + 1)2. Thus, it is of the form Identity I were a = x and b = 1. So we have,
(x + 1)2 = (x)2 + 2(x)(1) + (1)2 = x2 + 2x + 1
Example 2:
Factorise (x4 – 1) using standard algebraic identities.
Solution:
(x4 – 1) is of the form Identity III where a = x2 and b = 1. So we have,
(x4 – 1) = ((x2)2– 12) = (x2 + 1)(x2 – 1)
The factor (x2 – 1) can be further factorized using the same Identity III where a = x and b = 1. So,
(x4 – 1) = (x2 + 1)((x)2 –(1)2) = (x2 + 1)(x + 1)(x – 1)
Example 3:
Factorise 16×2 + 4y2 + 9z2 – 16xy + 12yz – 24zx using standard algebraic identities.
Solution:
16×2 + 4y2 + 9z2– 16xy + 12yz – 24zx is of the form Identity V. So we have,
16×2 + 4y2 + 9z2 – 16xy + 12yz – 24zx = (4x)2 + (-2y)2 + (-3z)2 + 2(4x)(-2y) + 2(-2y)(-3z) + 2(-3z)(4x)= (4x – 2y – 3z)2 = (4x – 2y – 3z)(4x – 2y – 3z)
Example 4:
Expand (3x – 4y)3 using standard algebraic identities.
Solution:
(3x– 4y)3 is of the form Identity VII where a = 3x and b = 4y. So we have,
(3x – 4y)3 = (3x)3 – (4y)3– 3(3x)(4y)(3x – 4y) = 27×3 – 64y3 – 108x2y + 144xy2
Example 5:
Factorize (x3 + 8y3 + 27z3 – 18xyz) using standard algebraic identities.
Solution:
(x3 + 8y3 + 27z3 – 18xyz)is of the form Identity VIII where a = x, b = 2y, and c = 3z. So we have,
(x3 + 8y3 + 27z3 – 18xyz) = (x)3 + (2y)3 + (3z)3 – 3(x)(2y)(3z)= (x + 2y + 3z)(x2 + 4y2 + 9z2 – 2xy – 6yz – 3zx)
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Frequently Asked Questions on Algebraic Identities
Q 1:- What are the three algebraic identities in Maths?
Ans:-The three algebraic identities in Maths are:
Identity 1: (a+b)^2 = a^2 + b^2 + 2ab
Identity 2: (a-b)^2 = a^2 + b^2 – 2ab
Identity 3: a^2 – b^2 = (a+b) (a-b)
Q 2:-What is the difference between an algebraic expression and identity?
Ans:- An algebraic expression is an expression that consists of variables and constants. In expressions, a variable can take any value. Thus, the expression value can change if the variable values are changed. But algebraic identity is equality which is true for all the values of the variables.
Q 3:- How to verify algebraic identity?
Ans:- The algebraic identities are verified using the substitution method. In this method, substitute the values for the variables and perform the arithmetic operation. Another method to verify the algebraic identity is the activity method. In this method, you would need a prerequisite knowledge of Geometry and some materials are needed to prove the identity.
