Identitate nabarmenak (baita Identitate nabariak edo Biderkadura nabarmenak ere) eragiketak egiteko askotan erabiltzen diren identitateak dira. Kalkulu aljebraikoan, zenbait adierazpen aljebraiko maiz agertzen dira, eta daukaten garrantziagatik identitate nabarmenak deritzegu.
Binomio baten berbidura ( a + b ) 2 = a 2 + 2 a b + b 2 {\displaystyle (a+b)^{2}=a^{2}+2ab+b^{2}\,} ( a − b ) 2 = a 2 − 2 a b + b 2 {\displaystyle (a-b)^{2}=a^{2}-2ab+b^{2}\,} ( 4 x 5 y + z ) 2 = 16 x 2 25 y 2 + 8 x z 5 y + z 2 {\displaystyle \left({\frac {4x}{5y}}+z\right)^{2}={\frac {16x^{2}}{25y^{2}}}+{\frac {8xz}{5y}}+z^{2}} ( 8 x + a ) 2 = 64 x 2 + 16 a x + a 2 {\displaystyle (8x+a)^{2}=64x^{2}+16ax+a^{2}\,} ( x − y ) 2 = ( x − y ) . ( x − y ) = x 2 − x y − y x + y 2 = x 2 − 2 x y + y 2 {\displaystyle (x-y)^{2}=(x-y).(x-y)=x^{2}-xy-yx+y^{2}=x^{2}-2xy+y^{2}\,} ( 3 m 4 n − p ) 2 = 9 m 2 16 n 2 − 6 m p 4 n + p 2 {\displaystyle \left({\frac {3m}{4n}}-p\right)^{2}={\frac {9m^{2}}{16n^{2}}}-{\frac {6mp}{4n}}+p^{2}} ( 1 − 2 x ) 2 = 1 − 4 x + 4 x 2 {\displaystyle (1-2x)^{2}=1-4x+4x^{2}\,} Binomio konjugatuak ( a + b ) . ( a − b ) = a 2 − a b + b a − b 2 = a 2 − b 2 {\displaystyle (a+b).(a-b)=a^{2}-ab+ba-b^{2}=a^{2}-b^{2}\,} ( a 2 + b 3 ) . ( a 2 − b 3 ) = a 4 − b 6 {\displaystyle (a^{2}+b^{3}).(a^{2}-b^{3})=a^{4}-b^{6}\,} ( a x − 2 ) . ( a x + 2 ) = a 2 x 2 − 4 {\displaystyle \left({\frac {a}{x}}-2\right).\left({\frac {a}{x}}+2\right)={\frac {a^{2}}{x^{2}}}-4} Binomio baten kuboa Binomio baten kuboaren bolumetria-deskonposizioa ( x + y ) 3 = x 3 + 3 x 2 y + 3 x y 2 + y 3 {\displaystyle (x+y)^{3}=x^{3}+3x^{2}y+3xy^{2}+y^{3}\,} ( x − y ) 3 = x 3 − 3 x 2 y + 3 x y 2 − y 3 {\displaystyle (x-y)^{3}=x^{3}-3x^{2}y+3xy^{2}-y^{3}\,} ( m + 3 n ) 3 = m 3 + 9 m 2 n + 27 m n 2 + 27 n 3 {\displaystyle (m+3n)^{3}=m^{3}+9m^{2}n+27mn^{2}+27n^{3}\,} ( x + 2 ) 3 = x 3 + 6 x 2 + 12 x + 8 {\displaystyle (x+2)^{3}=x^{3}+6x^{2}+12x+8\,} ( b − 2 c ) 3 = b 3 − 6 b 2 c + 12 b c 2 − 8 c 3 {\displaystyle (b-2c)^{3}=b^{3}-6b^{2}c+12bc^{2}-8c^{3}\,} ( x y − a b ) 3 = x 3 y 3 − 3 a x 2 b y 2 + 3 a 2 x b 2 y − a 3 b 3 {\displaystyle \left({\frac {x}{y}}-{\frac {a}{b}}\right)^{3}={\frac {x^{3}}{y^{3}}}-{\frac {3ax^{2}}{by^{2}}}+{\frac {3a^{2}x}{b^{2}y}}-{\frac {a^{3}}{b^{3}}}\,} ( 1 − x ) 3 = 1 − 3 x + 3 x 2 − x 3 {\displaystyle (1-x)^{3}=1-3x+3x^{2}-x^{3}\,} Trinomio baten berbidura ( a + b + c ) 2 = a 2 + b 2 + c 2 + 2 a b + 2 a c + 2 b c {\displaystyle (a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2ab+2ac+2bc\,} ( x + y + z ) 2 = x 2 + y 2 + z 2 + 2 x y + 2 x z + 2 y z {\displaystyle (x+y+z)^{2}=x^{2}+y^{2}+z^{2}+2xy+2xz+2yz\,} ( x − 2 y − 3 ) 2 = x 2 + ( − 2 y ) 2 + ( − 3 ) 2 + 2 x ( − 2 y ) + 2 x ( − 3 ) + 2 ( − 2 y ) ( − 3 ) {\displaystyle (x-2y-3)^{2}=x^{2}+(-2y)^{2}+(-3)^{2}+2x(-2y)+2x(-3)+2(-2y)(-3)\,} = x 2 + 4 y 2 + 9 − 4 x y − 6 x + 12 y {\displaystyle =x^{2}+4y^{2}+9-4xy-6x+12y\,} Stevinen biderkadura Gai komun bat duten 2 binomioen biderkadura.
( x + a ) . ( x + b ) = x 2 + ( a + b ) x + a b {\displaystyle (x+a).(x+b)=x^{2}+(a+b)x+ab\,} ( x + 4 ) ( x + 3 ) = x 2 + ( 4 + 3 ) x + 4.3 = x 2 + 7 x + 12 {\displaystyle (x+4)(x+3)=x^{2}+(4+3)x+4.3=x^{2}+7x+12\,} ( x − 2 ) ( x − 6 ) = x 2 + ( − 2 − 6 ) x + ( − 2 ) ( − 6 ) = x 2 − 8 x + 12 {\displaystyle (x-2)(x-6)=x^{2}+(-2-6)x+(-2)(-6)=x^{2}-8x+12\,} ( x − 1 ) ( x + 5 ) = x 2 + ( − 1 + 5 ) x + 5 ( − 1 ) = x 2 + 4 x − 5 {\displaystyle (x-1)(x+5)=x^{2}+(-1+5)x+5(-1)=x^{2}+4x-5\,} Warringen biderkadura ( a + b ) . ( a 2 − a b + b 2 ) = a 3 + b 3 {\displaystyle (a+b).(a^{2}-ab+b^{2})=a^{3}+b^{3}\,} ( a − b ) . ( a 2 + a b + b 2 ) = a 3 − b 3 {\displaystyle (a-b).(a^{2}+ab+b^{2})=a^{3}-b^{3}\,} ( x + 5 ) ( x 2 − 5 x + 25 ) = x 3 + 5 3 = x 3 + 125 {\displaystyle (x+5)(x^{2}-5x+25)=x^{3}+5^{3}=x^{3}+125\,} ( x − 3 ) . ( x 2 + 3 x + 9 ) = x 3 − 3 3 = x 3 − 27 {\displaystyle (x-3).(x^{2}+3x+9)=x^{3}-3^{3}=x^{3}-27\,} Arganden identitatea ( x 2 + x + 1 ) ( x 2 − x + 1 ) = x 4 + x 2 + 1 {\displaystyle (x^{2}+x+1)(x^{2}-x+1)=x^{4}+x^{2}+1\,} Gaussen identitateak a 3 + b 3 + c 3 − 3 a b c = ( a + b + c ) ( a 2 + b 2 + c 2 − a b − b c − a c ) {\displaystyle a^{3}+b^{3}+c^{3}-3abc=(a+b+c)(a^{2}+b^{2}+c^{2}-ab-bc-ac)\,} a 3 + b 3 + c 3 − 3 a b c = 1 2 ( a + b + c ) [ ( a − b ) 2 + ( b − c ) 2 + ( a − c ) 2 ] {\displaystyle a^{3}+b^{3}+c^{3}-3abc={\frac {1}{2}}(a+b+c)[(a-b)^{2}+(b-c)^{2}+(a-c)^{2}]\,} Legendreren identitateak ( a + b ) 2 + ( a − b ) 2 = 2 ( a 2 + b 2 ) {\displaystyle (a+b)^{2}+(a-b)^{2}=2(a^{2}+b^{2})\,} ( a + b ) 2 − ( a − b ) 2 = 4 a b {\displaystyle (a+b)^{2}-(a-b)^{2}=4ab\,} ( a + b ) 4 − ( a − b ) 4 = 8 a b ( a 2 + b 2 ) {\displaystyle (a+b)^{4}-(a-b)^{4}=8ab(a^{2}+b^{2})\,} Lagrangeren identitateak ( a 2 + b 2 ) ( x 2 + y 2 ) = ( a x + b y ) 2 + ( a y − b x ) 2 {\displaystyle (a^{2}+b^{2})(x^{2}+y^{2})=(ax+by)^{2}+(ay-bx)^{2}\,} ( a 2 + b 2 + c 2 ) ( x 2 + y 2 + z 2 ) = ( a x + b y + c z ) 2 + ( a y − b x ) 2 + ( a z − c x ) 2 + ( b z − c y ) 2 {\displaystyle (a^{2}+b^{2}+c^{2})(x^{2}+y^{2}+z^{2})=(ax+by+cz)^{2}+(ay-bx)^{2}+(az-cx)^{2}+(bz-cy)^{2}\,} Ikus, gainera Binomioa Pascalen hirukia Kanpo estekak
![{\displaystyle a^{3}+b^{3}+c^{3}-3abc={\frac {1}{2}}(a+b+c)[(a-b)^{2}+(b-c)^{2}+(a-c)^{2}]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c861b04d0eb0a8d7e5f8bec4a258a7d9665049b4)
Datuak: Q1971429
