Standard Identities

Algebra is a very important mathematical concept. It is applied not only in mathematics but in real life too. Standard and algebraic identities are used to simplify and restructure these complex problems and solve them with ease. The standard identities are identities that are obtained by the multiplication of binomials. And the algebraic identities can be referred to as the algebraic equation where LHS and RHS remain equal for every value of Variables in the equation. The importance of Standard identities is because of their unique properties of standard identities. The algebraic identities can also be verified in two ways, i.e., activity and substitution.

What are Standard Identities?

Identity can be defined as an equality relation, which stands true irrespective of the variable’s value put in the equation. In simpler terms, for being an identity, the left-hand side of the equation should be equal to the right-hand side of the equation, no matter what the variable’s value is used in the equation. The standard identities are identities obtained by multiplying binomials, a two term polynomial.

What are algebraic Identities?

Algebraic standard Identities can be defined as the algebraic equations that stand equal irrespective of the variable’s value. These identities are used to simplify or rearrange the algebra expressions. Both sides of the equation are interchangeable and replaceable. It can be inferred that not all equations are algebraic identities, but all algebraic identities are equations. In simpler words, not all algebraic equations are algebraic identities.

The important standard Algebraic equations are as follows-

1. (a+b)²=a²+2ab+b²

2. (a–b)²=a²–2ab+b²

3. (a+b)(a–b)=a²–b² 

4. (x+a)(x+b)=x²+(a+b)x+ab

5.  (a+b+c)2=a²+b²+c²+2ab+2bc+2ac

6. (a+b)³=a³+b³+3ab(a+b)

7. (a–b)³=a³–b³–3ab(a–b)

8. (a+b+c)(a²+b²+c²–ab–bc–ca)=a³+b³+c³–3abc

9. (a + b) (a + c) (b + c) = (a + b + c) (ab + ac + bc) – abc

10. a² + b² +c² = (a + b + c)² – 2(ab + ac + bc)

What is the Importance of Algebraic identity?

The standard Identities and Algebraic Identities are vital concepts of mathematics. They are used not only to simplify, rearrange and solve mathematical problems such as factorization, integration, trigonometry, and differentiation problems but also have life implications. This is done because of the unique properties of standard identities. They are true for any variable value, and one side can be used with another. They also help in solving complex equations in less time.

How do we verify Algebraic identities?

Verification of algebraic identity means-testing these identities. This can be done in two ways: the Activity Method and the Substitution Method. Both are discussed below;

• Activity Method – In this method, verification is done physically by an activity. Papers are cut according to the different values of x and y and pasted to understand the correlation between LHS and RHS. 

• Substitution Method- The arithmetic operations are used to verify the identities in this method. Different values of x and y are taken to prove that in every case, LHS is equal to RHS.

How do we differentiate algebraic expressions and algebraic identity?

An expression made up of constants and variables is an algebraic expression. At the same time, an algebraic identity is an algebraic equation that stands equal irrespective of the variable’s value. The identity value remains the same irrespective of the variable’s value used. However, the value of the expression changes as the value of the variable changes.

For example,

1. In the expression 10x – 3 = y, the value of y will change as the value of x changes. Both sides, i.e., RHS and LHS, will not be the same for every value of x and y.

2. But in the algebraic equation, (a-b)²= a² + b² – 2ab , 

                                                Suppose, a = 2 and b = 1

                                              = (2-1)²= 2² + 1² – 2×1×2

                                              = 1= 4+1-4

                                                          = 1= 1

                                         Therefore, LHS = RHS.

For every value of a and b this equation will be true in this algebraic identity. 

Conclusion 

Identities lay down the pathway to understanding what should be the approach to a mathematical problem. The properties of standard identities help build a deeper understanding and simplify complex mathematical problems. Different standard identities are used for different mathematical concepts. The algebraic identities are widely used to solve algebraic equations with ease. It is a proven formula used in solving equations with a higher power. It understands advanced mathematical concepts such as integration, differentiation, trigonometry, and many more.