Productos notables 5º
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PRODUCTOS NOTABLES
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BINOMIO AL CUADRADO
EQUIVALENCIA DE LEGENDRE
2 2 2
2 2 2
( ) 2
( ) 2
a b a ab b
a b a ab b
2 2 2 2
2 2
( ) ( ) 2( )
( ) ( ) 4
a b a b a b
a b a b ab
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DIFERENCIA DE CUADRADOS
TRINOMIO AL CUADRADO
2 2( )( )a b a b a b
2 2 2 2( ) 2 2 2a b c a b c ab ac bc
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BINOMIO AL CUBO
EQUIVALENCIA DE CAUCHY
3 3 2 2 3
3 3 2 2 3
( ) 3 3
( ) 3 3
a b a a b ab b
a b a a b ab b
3 3 3
3 3 3
( ) 3( )( )
( ) 3( )( )
a b a b ab a b
a b a b ab a b
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SUMA Y DIFERENCIA DE CUBOS
TRINOMIO AL CUBO
3 3 2 2
3 3 2 2
( )( )
( )( )
a b a b a ab b
a b a b a ab b
3 3 3 3( ) 3( )( )( )a b c a b c a b a c b c
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EQUIVALENCIA CONDICIONALSi los números a, b y c. Se verifican: a + b + c = 0
2 2 2
2 2 2 2
3 3 3
4 4 4 2 2 2 2
5 5 5
2( )
( ) ( ) ( ) ( )
31 ( )25 ( )
a b c ab ac bc
ab ac bc ab ac bc
a b c abc
a b c a b c
a b c abc ab ac bc
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IMPLICANCIA NOTABLESiendo a, b y c números reales tenemos:
2 2 2
3 3 3 3 0
a b c ab ac bc a b c
a b c abc a b c a b c
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EJEMPLO Nº 01Si:
Calcula:
3 3 5 ..........(1)( ) 1 ..........(2)x yxy x y
2( )x y
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EJEMPLO Nº 02Si:
Calcula:
2 ..........(1)3 ..........(2)
a bab
2 2 3 3M a b a b
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PROBLEMA Nº 01Si:
Calcula:
1 5x x
3 3x x
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PROBLEMA Nº 02Si:
Calcula:
1 1 4x y x y
53x yx y
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PROBLEMA Nº 03Si:
Calcula:
2 2 2
57
a b ca b c
ab ac bc
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PROBLEMA Nº 04Simplifica:
( )( )x a b x a c bcEx a b c
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PROBLEMA Nº 05Si:
Calcula:
2 2 2 ; , ,a b c ab ac bc a b c
5 3n n n
n
a b cKb
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PROBLEMA Nº 06Si:
Calcula:
2 2 8x yz x yz yz
2 2x yz x yz
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PROBLEMA Nº 07Si:
Calcula:
5 3 2
2 3 2 5
5 2 3
a
b
c
3 3 3 2 2 2( )( )( )
a b c a b cKabc ab ac bc
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PROBLEMA Nº 01Si:
Calcula:
3 3
2 2 3
100 10 1
10 1
ab
a b
4 4( ) ( )a b a b
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PROBLEMA Nº 02Si:
Calcula:
3 33 14 3 141 15 5 5 5
x
35 3 1x x
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PROBLEMA Nº 03Si:
Calcula:
3 3 3
2 2 2
32
a b ca b c
( )(2 )1
a b c ab ac bcFabc
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PROBLEMA Nº 04Simplifica:
2 2 27 5 5 2 2 2 2 7
7 5 5 2 2 5 2 2 2 2 7 7 5 2 2 7
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PROBLEMA Nº 05Si x, y R tal que:
Calcula el mayor valor que puede asumir «x + y»:
2 2 8x y