ALGEBRAIC IDENTITIES
Algebra is one of the most important chapters of basic mathematics. Students get to know about Algebraic Identities in the lower grades, at the high school level, and then move up to the upper grades and learn higher levels of algebraic Identities. Algebraic identification is a broad topic and is useful in all areas of a student's life. An algebraic identifier is an algebraic equation that applies to all variable values in it. An algebraic equation is a mathematical expression consisting of numbers, variables (unknown values), and mathematical functions (addition, subtraction, multiplication, division, etc.) They are mainly used to find elements of polynomials.
Everything About Algebraic Identities If the equation is true for all the values of the variables in it, it is called an identifier. An algebraic identifier is an equation where the value of the left-hand side of the equation is equal to the value of the right-hand side of the equation for all variable values. We have several standard identifiers that we can use in different branches of mathematics. All standard Identities are obtained by the Binomial statement. An algebraic equation that refers to all the values of a variable in it is called an algebraic identifier. It is also used to factor polynomials. Thus, algebraic identifiers are used in the calculation of algebraic expressions and in the solution of various polynomials.
Standard Algebraic Identities List
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
where 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 factorised 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 16x² + 4y² + 9z² – 16xy + 12yz – 24zx using standard algebraic identities.
Solution:
16x2 + 4y2 + 9z2– 16xy + 12yz – 24zx is of the form Identity V.
So we have,
16x2 + 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)3using 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)
= 27x3 – 64y3 – 108x2y + 144xy2
Example 5:
Factorise (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)2 + (3z)3 – 3(x)(2y)(3z)
= (x + 2y + 3z)(x2 + 4y2 + 9z2 – 2xy – 6yz – 3zx)
What is the difference between an algebraic expression and identities?
An algebraic expression is an expression which 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.
How to verify algebraic identity?
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.
What is the use of algebraic identities?
Algebraic identities are used to solve the algebraic expression or polynomial faster. It makes the calculation easier.