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( ) 2 2 2 36) x+ 3 = x + 2⋅x⋅ 3 + 3 = x 2 + 6x+ 9 100 Elementarnih zadataka ( ) ( ) ( ) 2 2 2 2 2 2 2 37 ) 2x+ 3y = 2x + 2⋅2x⋅ 3y+ 3y = 2 x + 12xy+ 3 y = 4x + 12xy+ 9y ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 38) 3x− 4y = 3x −2⋅3x⋅ 4y+ 4y = 3 x − 24xy+ 4 y = 9x − 24xy+ 16y 2 2 2 2 2 2 3 3 3 2 3 2 ⎛1 2 ⎞ ⎛1 ⎞ 1 2 ⎛2 ⎞ 1 2 2 39) ⎜ x+ y ⎟ = ⎜ x⎟ + 2⋅ x⋅ y + ⎜ y ⎟ = x + xy + ⎝2 3 ⎠ ⎝2 ⎠ 2 3 ⎝3 ⎠ 2 3 3 2 2 ( y ) 1 2 4 = x + xy + y 4 3 9 2 3 2 3 6 2 2 2 2 2 2 2 3 4 3 3 4 4 3 223 ⋅ ⋅ 3 4 3 2 ( ) 2 ( ) 40) ⎛2 3 ⎞ ⎛2 ⎞ 2 3 ⎛3 ⎞ 2 3 ⎜ x − y ⎟ = ⎜ x ⎟ −2⋅ x ⋅ y + ⎜ y ⎟ = x − ⋅x ⋅ y + ⎝3 4 ⎠ ⎝3 ⎠ 3 4 ⎝4 ⎠ 3 3⋅4 4 x = 4 6 3 4 9 8 = x − x y + y 9 16 41) −2 1 1 1 ( 5x− 6y) = = = 2 2 2 2 2 25x − 60xy+ 36y ( 5x−6y) ( 5x) −2⋅5x⋅ 6y+ ( 6y) 2 2 ( − a− y) = ( − ⋅ ( a+ y) ) = ( − ) 42) 2 3 1 2 3 1 ( ) ( ) ( ) 2 2 2 2 43) 2x−5 ⋅ 2x+ 5 = 2x − 5 = 2 x − 25 = ( ) ( 2a 3y) 1 ( 2a) 2 2a 3y ( 3y) 2 2 2 2 ⋅ + = ⋅ + ⋅ ⋅ + = 4a + 12ay+ 9y 2 4 25 x − 2 2 44) 2 2 2 2 ⎛2 2 2 2 4 3⎞ ⎛2 2 4 3⎞ ⎛2 2⎞ ⎛4 3⎞ 2 2 4 3 4 16 ⎜ x − y ⎟⋅ ⎜ x + y ⎟ = ⎜ x ⎟ − ⎜ y ⎟ = 2 ( x ) − 2 ( y ) = x − y ⎝3 5 ⎠ ⎝3 5 ⎠ ⎝3 ⎠ ⎝5 ⎠ 3 5 9 25 4 6 2 2 2 3 4 5 6 2 3 4 5 6 2 3 4 5 6 ( x y z − a b ) ⋅( x y z − a b ) = ( x y z ) −( ab ) = 2 2 3 4 2 5 6 3 ( x ) ( y ) ( z ) 4 ( a ) ( b ) 45) 3 4 3 4 3 4 ( ) 3 3 2 2 3 3 2 + ( ) ( ) ( ) ( ) ( ) ( ) ( y z) ( y) ( y) z y ( z) 2 2 2 2 2 46) x+ 3 = x + 3⋅x ⋅ 3 + 3⋅x⋅ 3 + 3 = x + 9x + 3⋅9⋅ x+ 27 = 2 2 1 2 1 3 3 2 2 3 3 3 2 2 = − = 9x y z −16a b + 9 + 27 + 27 3 2 x x x 4 6 8 10 12 47 ) x+ 2 ⋅ x+ 2 = x+ 2 ⋅ x+ 2 = x+ 2 = x+ 2 = x + 3⋅x ⋅ 2 + 3⋅x⋅ 2 + 2 = ( ) 3 3 3 2 2 2 2 3 3 48) 2 − 3 = 2 −3⋅ 2 ⋅ 3 + 3⋅2 ⋅ 3 − 3z = 2 y −9⋅2 ⋅y ⋅ z+ 6⋅y⋅3 ⋅z − 3 z = = 8y − 36y z+ 54yz −27z 3 2 2 3 + 6 + 12 + 8 3 2 x x x 3 3 2 2 3 3 3 2 3 3 3 49) ⎛1 1 ⎞ ⎛1⎞ ⎛1⎞ 1 1 ⎛1 ⎞ ⎛1 ⎞ 1 1 1 1 1 1 ⎜ + x⎟ = ⎜ ⎟ + 3⋅⎜ ⎟ ⋅ x+ 3⋅ ⋅ ⎜ x⎟ + ⎜ x⎟ = + 3⋅ ⋅ ⋅ x+ 3⋅ ⋅ ⋅ x + ⎝2 3 ⎠ ⎝2⎠ ⎝2⎠ 3 2 ⎝3 ⎠ ⎝3 ⎠ 2 4 3 2 9 3 x = 1 1 1 2 1 3 = + x+ x + x 8 4 6 27 50) 3 3 2 2 3 2 3 2 ⎛2 3 ⎞ ⎛2 ⎞ ⎛2 2⎞ 3 3 2 2 ⎛3 3⎞ ⎛3 3⎞ ⎜ x − y ⎟ = x 3 x y 3 x y y ⎝3 2 ⎜ ⎠ 3 ⎟ − ⋅⎜ 3 ⎟ ⋅ + ⋅ ⋅ − = 2 3 ⎜ 2 ⎟ ⎜ 2 ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 3 2 2 3 2 3 2 2 3 2 2 2 3 3 2 3 3 3 3 = ( x 3 ) −3⋅ 2 ( x ) ⋅ y + 2⋅x ⋅ 2 ( y ) − 3 ( y ) = 3 3 2 2 2 8 6 4 3 9 2 6 27 9 = x − 2x y + x y − y 27 2 8 3 3 2 2 3 2 3 4 2 3 2 3 4 2 3 4 4 ( x y − z ) = ( x y ) − ⋅( x y ) ⋅ z + ⋅ x y ⋅( z ) − ( z ) = 3 3 2 3 2 3 2 3 4 2 3 4 3 4 = 3 ( x ) ( y ) −3⋅2⋅9⋅( x ) ⋅( y ) z 9x y 4 ( z ) 2 ( z ) 51) 3 2 3 3 3 2 3 3 2 2 = 27x y − 54x y + 36x y z −8z 6 9 4 6 2 3 8 12 2 2 3 ⋅ + ⋅ ⋅ − = 3 3 3 2 2 3 3 2 2 3 ( x y) ( x y) x x y xy y ( x x y xy y ) 52) + − − = + 3 + 3 + − − 3 + 3 − = x 3x y 3xy y x 3x y 3xy y 3 2 2 3 3 2 2 3 = + + + − + − + = 6x y+ 2y 2 3 © M I M SRAGA 1999 / 2004 14
100 Elementarnih zadataka © M I M SRAGA 1999 / 2004 15
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