\(\left(x+y\right)^3-\left(x-y\right)^3-2y^3\)
\(=x^3+y^3+3x^2y+3xy^2-x^3+y^3+3x^2y-3xy^2-2y^3\)
\(=2y^3+6x^2y-2y^3\)
\(=6x^2y\)
Với a+b+c=0 thì \(a^3+b^3+c^3=a^3+b^3-\left(a+b\right)^3=a^3+b^3-a^3-b^3-3ab\left(a+b\right)\)
\(=-3ab\left(a+b\right)=-3ab.\left(-c\right)=3abc\)
Áp dụng:\(\left(x+y\right)^3-\left(x-y\right)^3-\left(2y\right)^3=\left(x+y\right)^3+\left(y-x\right)+\left(-2y\right)^3\)
có \(x+y+y-x+\left(-2y\right)=0\)\(\Rightarrow\left(x+y\right)^3-\left(x-y\right)^3-\left(2y\right)^3=3.\left(-2y\right).\left(x+y\right)\left(y-x\right)\)
\(=6y\left(x+y\right)\left(x-y\right)\)\(=6y\left(x^2-y^2\right)=6x^2y-6y^3\)
Với \(a+b+c=0\)thì \(a^3+b^3+c^3\)
\(=a^3+b^3-\left(a+b\right)^3\)
\(=a^3+b^3-a^3-b^3-3ab\left(a+b\right)\)
\(=-3ab\left(a+b\right)\)
\(=-3ab.\left(-c\right)\)
\(=3abc\)
Áp dụng: \(\left(x+y\right)^3-\left(x-y\right)^3-\left(2y\right)^3\)
\(=\left(x+y\right)^3+\left(y-x\right)+\left(-2y\right)^3\)
Có: \(x+y+y-x+\left(-2y\right)=0\)
\(\Rightarrow\left(x+y\right)^3-\left(x-y\right)^3-\left(2y\right)^3\)
\(\Rightarrow3.\left(-2y\right).\left(x+y\right)\left(y-x\right)\)
\(\Rightarrow6y\left(x+y\right)\left(x-y\right)\)
\(\Rightarrow6y\left(x^2-y^2\right)\)
\(\Rightarrow6x^2y-6y^3\)
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