# CLASS-7SOME SPECIAL RULES OF MULTIPLICATION OR PRODUCT

SOME SPECIAL RULES OF MULTIPLICATION OR PRODUCT

There are some special important rules of multiplication are to be discussed here. These special products come in handy while working out a problem and very useful to remember them –

1)  ( X + a ) ( X + b )  =  X² + ( a + b ) X + ab

( X + a ) ( X + b ) =  ( X + a ) . X  + ( X + a ) . b

[ By the Distribution Law ]

= X² + a X + b X + ab  [ By the Distributive Law ]

=  X² + ( a + b ) X + ab

[ a X = b X , by the commutative law ]

2)  ( a + b )²  =   a² + 2ab + b²

( a + b )² =  ( a + b ) . ( a + b )        [ By the Distribution Law ]

=   a ( a + b ) + b ( a + b )   [ By the Distribution Law ]

=  a² + ab + ba + b²  [ a b = b a, by the commutative law ]

=   a² + 2ab + b²

( Sum of two terms )² = ( first term )² + 2 X first term X second term + ( second term )²

3)  ( a - b )² =  a² - 2ab + b²

( a - b )² =   ( a - b ) . ( a - b )    [ By the Distribution Law ]

= a ( a - b ) - b ( a - b )  [ By the Distribution Law ]

=  a² - ab - ba + b²  [a b = b a, by the commutative law]

=   a² - 2ab + b²

( Difference of two terms )² = ( first term )² - 2 X first term X second term + ( second term )²

4)  ( a + b ) ( a – b )  = a² - b²

( a + b ) ( a – b ) = a ( a – b ) + b ( a – b ) [ By the Distribution Law ]

=  a² - ab + ba - b²     [ By the Distribution Law ]

=   a² - b²   [ a b = b a , by the commutative law ]

( first term + second term )  ( first term – second term ) = ( first term )² - (second term )²

5)  a² + b² = ( a + b )² - 2ab           or

a² + b² = ( a - b )² + 2ab

6)  ( X - a ) ( X - b )  =  X² - ( a + b ) X + ab

( X - a ) ( X - b )  =  ( X - a ) . X  - ( X - a ) . b

[ By the Distribution Law ]

= X² - a X - b X + ab  [ By the Distributive Law ]

=  X² - ( a + b ) X + ab

[ a X = b X, by the commutative law ]

7)  Commutative Law

a + b = b + a

a . b = b . a

8) Distributive Law

a + b . c = ( a + b ) ( a + c )

a ( b + c ) = a . b + a . c

9) a + 0 = 0 + a = a

a . 1 = 1 . a = a