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