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Содда кунед:

\((x+y+z)^3-(x+y-z)^3-(x-y+z)^3-(-x+y+z)^3\)

a=x+y ва b=x-y

\((x+y+z)^3-(x+y-z)^3-(x-y+z)^3-(-x+y+z)^3=\)

\(=((a+z)^3-(a-z)^3)-((b+z)^3+(-b+z)^3)=\)

\(=((a+z)^3-(a-z)^3)-((b+z)^3-(b-z)^3)\)

\((a+z)^3-(a-z)^3=(a+z-a+z)\cdot((a+z)^2+(a+z)(a-z)+(a-z)^2)=\)

\(=2z\cdot(a^2+z^2+a^2-z^2+a^2+z^2)=2z\cdot(3a^2-z^2)\)

\((b+z)^3-(b-z)^3=2z\cdot(3b^2-z^2)\)

\(((a+z)^3-(a-z)^3)-((b+z)^3-(b-z)^3)=\)

\(=2z\cdot(3a^2-z^2)-2z\cdot(3b^2-z^2)=\)

\(=2z\cdot(3a^2-z^2-3b^2+z^2)=2z\cdot(3a^2-3b^2)=\)

\(=6z\cdot(a^2-b^2)\)

\(a^2=x^2+2xy+y^2\)

\(b^2=x^2-2xy+y^2\)

\(a^2-b^2=x^2+2xy+y^2-(x^2-2xy+y^2)=\)

\(=4xy\)

\(6z\cdot(a^2-b^2)=6z\cdot4xy=\)

\(=24xyz\)

\((x+y+z)^3-(x+y-z)^3-(x-y+z)^3-(-x+y+z)^3=24xyz\)

Ҷавоб: 24xyz.