Identitate nabarmen

identitate nabarmenak (edo biderkadura nabarmenak) eragiketak egiteko askotan erabiltzen diren identitateak dira. Kalkulu aljebraikoan, zenbait adierazpen aljebraiko maiz agertzen dira, eta daukaten garrantziagatik identitate nabarmenak deritzegu.

Binomio baten berbidura

$(a+b)^2= a^2+2ab+b^2 \,$ $(a - b)^2 = a^2 - 2 a b + b^2 \,$

• Adibideak:

1. $\left( \frac{4x}{5y}+z \right )^2=\frac{16x^2}{25y^2}+\frac{8xz}{5y}+z^2$
2. $(8x+a)^2=64x^2+16ax+a^2 \,$
3. $(x - y)^2 = (x -y) . (x - y) = x^2 - xy - yx + y^2 = x^2 - 2xy + y^2 \,$
4. $\left( \frac{3m}{4n}-p \right )^2=\frac{9m^2}{16n^2}-\frac{6mp}{4n}+p^2$
5. $(1-2x)^2=1-4x+4x^2 \,$

Binomio konjugatuak

$(a + b).(a - b) = a^2 - ab + ba - b^2=a^2-b^2 \,$

• Adibideak:

1. $(a^2+b^3).(a^2-b^3)=a^4-b^6 \,$
2. $\left( \frac{a}{x}-2 \right ).\left( \frac{a}{x}+2 \right )=\frac{a^2}{x^2}-4$

Binomio baten kuboa

Binomio baten kuboaren bolumetria-deskonposizioa

$(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3 \,$ $(x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3 \,$

• Adibideak:

1. $(m+3n)^3=m^3+9m^2n+27mn^2+27n^3 \,$
2. $(x+2)^3=x^3+6x^2+12x+8 \,$
3. $(b-2c)^3=b^3-6b^2c+12bc^2-8c^3\,$
4. $\left ( \frac{x}{y}-\frac{a}{b} \right )^3=\frac{x^3}{y^3}-\frac{3ax^2}{by^2}+\frac{3a^2x}{b^2y}-\frac{a^3}{b^3}\,$
5. $(1-x)^3=1-3x+3x^2-x^3\,$

Trinomio baten berbidura

$(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc \,$

• Adibideak:

1. $(x+y+z)^2=x^2+y^2+z^2+2xy+2xz+2yz \,$
2. $(x-2y-3)^2=x^2+(-2y)^2+(-3)^2+2x(-2y)+2x(-3)+2(-2y)(-3) \,$
   

$= x^2+4y^2+9-4xy-6x+12y \,$

Stevinen biderkadura

Gai komun bat duten 2 binomioen biderkadura.

$(x + a).(x + b) = x^2 + (a + b)x + ab \,$

• Adibideak:

1. $(x+4)(x+3)=x^2+(4+3)x+4.3=x^2+7x+12 \,$
2. $(x-2)(x-6)=x^2+(-2-6)x+(-2)(-6)=x^2-8x+12 \,$
3. $(x-1)(x+5)=x^2+(-1+5)x+5(-1)=x^2+4x-5 \,$

Warringen biderkadura

$(a + b).(a^2 - ab + b^2) = a^3 + b^3 \,$ $(a - b).(a^2 + ab + b^2) = a^3 - b^3 \,$

• Adibideak:

1. $(x+5)(x^2-5x+25)=x^3+5^3=x^3+125 \,$
2. $(x - 3).(x^2 + 3x + 9) = x^3 - 3^3 = x^3 - 27 \,$

Arganden identitatea

$(x^2+x+1)(x^2-x+1) = x^4+x^2+1 \,$

Gaussen identitateak

$a^3+b^3+c^3-3abc= (a+b+c)(a^2+b^2+c^2-ab-bc-ac) \,$ $a^3+b^3+c^3-3abc= \frac{1}{2} (a+b+c)[(a-b)^2+(b-c)^2+(a-c)^2] \,$

Legendreren identitateak

$(a+b)^2+(a-b)^2=2(a^2+b^2) \,$ $(a+b)^2-(a-b)^2=4ab \,$ $(a+b)^4-(a-b)^4=8ab(a^2+b^2) \,$

Lagrangeren identitateak

$(a^2+b^2)(x^2+y^2) = (ax+by)^2+(ay-bx)^2 \,$ $(a^2+b^2+c^2)(x^2+y^2+z^2) = (ax+by+cz)^2+(ay-bx)^2+(az-cx)^2+(bz-cy)^2 \,$

Ikus, gainera

• Binomioa
• Pascalen hirukia