## ALGEBRAIC IDENTITY

On this webpage algebraic identity, we are going to see the list of identities which are being used to solve all kind of problems in the topic Algebra in math.

## What is algebraic identity?

Algebraic identity is an equality which remains true regardless of the value of any variable which appear within it.

## Definition of formula

Formula is the short cut of doing problems without doing the complete steps. It is very long process of doing any problem without formula. Let us consider the following example to understand the purpose of formulas in math.

 Formulas Examples & Practice questions

 (1) (a + b)² = a² + 2 ab + b²(2) (a + b)² = (a - b)² + 4 ab Examplespractice questions (3) (a - b)² = a² - 2 ab + b²(4) (a - b)² = (a + b)² - 4 ab ExamplesPractice Questions (5) a² - b² = (a + b) (a - b) ExamplesPractice Questions (6) (x+a) (x+b) =x² + (a + b) x+a b ExamplesPractice Questions (7) (a+b)³=a³+3a²b+3ab²+b³(8) (a+b)³=a³+b³+3ab(a+b) ExamplesPractice Questions (9) (a-b)³=a³-3a²b+3ab²-b³(10) (a-b)³=a³-b³-3ab(a-b) ExamplesPractice Questions (11)  a³+b³ = (a+b)(a²-ab+b²)(12)  a³+b³=(a+b)³-3 ab(a + b) ExamplesPractice Questions (13) a³-b³= (a-b)(a²+ab+ b²)(14) a³-b³=(a-b)³ +3ab(a-b) ExamplesPractice Questions
 (15) (a+b+c)²= a²+b²+c² +2ab+2bc+2ca(16) (a+b-c)²=a²+b²+c² +2ab-2bc-2ca(17) (a-b+c)²= a²+b²+c²-2ab-2bc+2ca(18) (a-b-c)²= a²+b²+c²-2ab+2bc-2ca (19) a² + b² = (a + b)² - 2ab (20) a² + b² = (a - b)² + 2ab (21) a² +  b²=½ [(a+b)²-(a-b)²] (22) ab = ¼[(a+b)²- (a - b)²] ExamplesPractice Questions

(23) (a + b + c)³ = a³ + b³ + c³ + 3a²b + 3a²c + 3ab²                                          + 3b²c + 3ac² + 3bc² + 6abc

(24) (a + b - c)³ = a³ + b³ - c³ + 3a²b - 3a²c + 3ab²                                            - 3b²c + 3ac² + 3bc² - 6abc

(25) (a - b + c)³ = a³ - b³ + c³ - 3a²b + 3a²c + 3ab²                                            + 3b²c + 3ac² - 3bc² - 6abc

(26) (a - b - c)³ = a³ - b³ - c³ - 3a²b - 3a²c + 3ab²                                             - 3b²c + 3ac² - 3bc² + 6abc

## How to remember algebraic identity with negative sign?

We can remember the expansion of the identities like (a+b)² (a+b+c)², (a+b+c)³. In the above identities, if one or more terms is negative, how can we remember the expansion?

This question has been answered in the following two cases.

Case 1 :

For example, let us consider the identity of (a + b + c)²

We can easily remember the expansion of (a + b + c)².

If "c" is negative, we will have (a + b - c)²

How can we remember the expansion of (a + b - c)² ?

It is very simple.

In the terms of the expansion, a², b², c²,  ab, bc, ca, let us consider the terms in which we find "c"

They are c², bc, ca .

Even if we take negative sign for "c", the sign of  will be positive. Because it has even power "2"

The terms bc, ca will be negative, Because both "b" and "a" are multiplied by "c" which is negative.

Finally, we have

(a + b - c)²= a² + b² + c² + 2ab - 2bc - 2ca

Case 2 :

In (a+b+c)², if both "b" and "c" are negative, we will have (a - b - c)²

How can we remember the expansion of (a - b - c)² ?

It is very simple.

In the terms of the expansion, a², b², c²,  ab, bc, ca, let us consider the terms in which we find "b" and "c"

They are b², c², ab,  bc, ca.

Even if we take negative sign for "b" and "c", the sign of b² and  will be positive. Because they even power "2".

The terms "ab" and "ca" will be negative.

Because, in "ab", "a" is multiplied by "b" which is negative.

Because, in "ca", "a" is multiplied by "c" which is negative.

The term "bc" will be positive.

Because, in "bc", both "b" and "c" are negative.                                           (negative x negative = positive)

Finally, we have

(a - b - c)²= a² + b² + c² - 2ab + 2bc - 2ca

In the same we can get idea to remember the the expansions of       (a + b - c)³, (a - b + c)³ (a - b - c)³

We hope that the students would have understood "How to remember algebraic identity with negative sign once they remember the expansions with positive sign".