Algebraic Identities Chart :

FACTORIZATION :

 Factorization, is breaking down a number into smaller numbers, that on multiplication gives the original number. The smaller numbers are called Factors or Divisors of that number.

Let the number be 12,

the number 12 is written as 2 times 6 = 2*6 , the factors are 2 and 6.

            (or)

 3 times 4 = 3*4 , the factors are 3and 4.

 (or)

 1 times 12 = 1*12, the factors are 1 and 12.

 

THE ALGEBRAIC IDENTITIES:

1. Square of a Binomial Identities:

i) (a + b)² = a² + 2 _* a _* b +  b² = a² + 2 ab + b² = (a + b)( a + b) .

                           (or)

    (- a – b )² = a² + 2 _* a _* b + b^2. = a²  + 2 ab + b²  =  (a + b) ( a + b).

ii) (a – b)² = a² – 2_* a _* b + b²  =  a² – 2ab + b²  = ( a – b ) ( a – b ).  

2. Difference of Squares Identity :

  a² – b² = (a – b )(a + b)

3. Cube of a Binomial Identity:

 (a + b)³  = a³ + 3a²b + 3ab² + b³.

 ( a – b )³ = a³ – 3a²b + 3ab² – b^3.

 

4. Sum of Cubes Identity:

a³ + b³  = (a + b) ( a²   - ab + b² ) 

              = (a + b)³ – 3ab( a + b )

5. Difference of Cubes Identity:

a³ – b³ = ( a – b )(a² + ab + b² ).

            = ( a – b )³ + 3ab ( a – b )

6. Product of Two Binomials Identity:

( x + a)( x + b ) = x² + (a + b) x + ab.

7. Squares of Trinomial Identities :

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

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

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

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

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

 

8. Cubes of Trinomial Identity :

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

 If  ( a + b + c ) = 0 ; then a³ + b³ + c³ = 3 abc .