Formulas of abridged multiplication
From the rules of multiplication of sums and
polynomials the following seven formulas of abridged multiplication can
be easily received. It is necessary
to know them by heart, as they are used in most of problems in mathematics.
[1] ( a + b )²
= a² + 2ab + b² ,
[2] ( a – b )²
= a² – 2ab + b²
,
[3] ( a + b ) ( a – b ) = a²
– b²,
[4] ( a + b )³
= a³
+ 3a² b + 3ab²
+ b³ ,
[5] ( a – b )³
= a ³
– 3a² b + 3ab²
– b³ ,
[6] ( a + b )( a²
– ab + b²
) = a³ + b³ ,
[7] ( a – b )( a ²
+ ab + b²
) = a³ – b³
.
E x a m p l e : Calculate 99³ using the formula [5] .
S o l u t i o n : 99³ = (100 – 1)³ = 1000000 – 3 ·10000 ·1 + 3 ·100 ·1 – 1 = 970299.
S o l u t i o n : 99³ = (100 – 1)³ = 1000000 – 3 ·10000 ·1 + 3 ·100 ·1 – 1 = 970299.
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