Page 6 - (1-1)eng

Elly fat mn 3omry ( 1 – 1 )
Some Identities

2
2
2
2
2
2
 ( a + b ) = a + 2 a b + b ……………………..……….. ( 2 x + 3 y ) = 4 x + 12 x y + 9 y
2
2
2
2
2
2
( a – b ) = a – 2 a b + b …………………..…….……… ( 2 x – 3 y ) = 4 x – 12 x y + 9 y
2
2
2
 ( a + b ) ( a – b ) = a – b ……………….……..…………………... ( x – 4 ) ( x + 4 ) = x – 16
3
3
2
2
 ( a – b ) ( a + a b + b ) = a – b ……………………....  x  2  x  2 3 x  4   x 
2
3
8
3
2
2

3
3
3
3
2
( a + b ) ( a – a b + b ) = a + b ……………………..……  x  1  x  3 x   1  x 1

Factorization
2
 When factorizing, first look for a common factor: x – 4 x = x ( x – 4 )
 To factorize an expression containing:

2 terms 3 terms 4 terms
 Difference of two squares  Perfect square  Grouping
2
2
2
2
x – 3 x – x + 3
a – b = ( a – b ) ( a + b ) a ± 2 ab + b = ( a ± b ) 2 3 2
2
2
2
3
4 x – 9 = ( 2 x – 3 ) ( 2 x + 3 ) 9 x + 12 x + 4 = ( 3 x +2 ) 2 = (x – 3 x ) – (x – 3)
2
2
x – 1 =  x  1   x  1  x – 2 x + 1 = ( x – 1 ) 2 = x (x – 3) – (x – 3)
2
= (x – 3) (x – 1)
= (x – 3) (x – 1) (x + 1)
 Sum of (difference between) two cubes  Trial and Error
3
3
a  b = (a  b)( a  a b + b )
2
2
3
4
3
2
x – 27 = ( x – 3 ) ( x + 3 x + 9 ) 6 x +13 x + 6 = (2 x + 3) (3 x + 2) x – x + x – 1
2
3
2
3
x + 27 = ( x + 3 ) ( x – 3 x +9 ) 6 x – 13 x + 6 = (2 x – 3) (3 x – 2) = x (x – 1) + (x – 1)
2
2
3
3
3
2
x – 1 =  x  1  x  3 x   1 6 x – 5 x – 6 = (2 x – 3) (3 x + 2) = (x – 1) (x + 1)
3
2
3
2
x + 1 =  x  1  x  3 x   1 10 x + 3 x – 18 = (2 x + 3) (5 x – 6) =(x – 1) (x + 1) (x – x + 1)
2
Calculus 5 Unit (1)