ALGEBRAIC IDENTITIES EXPANSIONS
Expansion Calculator for (a+b) to the power "n"

About our algebraic identities expansions 

On this web page "Algebraic identities expansions", we are going to have expansion calculator for (a + b) to the power "n" and also the list of identities and their expansions which are being used to solve all kinds of problems in algebra.   


   

Result:

      Expansion of (a+b)n for the given value of "n"



(1) (a + b)² = a² + 2 ab + b²

(2) (a + b)² = (a - b)² + 4 ab

Examples

practice questions

(3) (a - b)² = a² - 2 ab + b²

(4) (a - b)² = (a + b)² - 4 ab

Examples

Practice Questions

(5) a² - b² = (a + b) (a - b)

Examples

Practice Questions

(6) (x+a) (x+b) =x² + (a + b) x+a b

Examples

Practice Questions

(7) (a+b)³=a³+3a²b+3ab²+b³

(8) (a+b)³=a³+b³+3ab(a+b)

Examples

Practice Questions

(9) (a-b)³=a³-3a²b+3ab²-b³

(10) (a-b)³=a³-b³-3ab(a-b)

Examples

Practice Questions

(11)  a³+b³ = (a+b)(a²-ab+b²)

(12)  a³+b³=(a+b)³-3 ab(a + b)

Examples

Practice Questions

(13) a³-b³= (a-b)(a²+ab+ b²)

(14) a³-b³=(a-b)³ +3ab(a-b)

Examples

Practice 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]

Examples

Practice 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 identities expansions with negative sign?

We can remember algebraic identities expansions 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 identities expansions with negative sign once they remember the expansions with positive sign". 

Apart from what has been given on this web page, if you want to know more about algebraic identities expansions, please click here. 

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