If (x + y)³ − (x − y)³ − 6y(x² − y²) = ky³, Find k
If \[ (x+y)^3-(x-y)^3-6y(x^2-y^2)=ky^3, \] then \[ k= \]
(a) \(1\)
(b) \(2\)
(c) \(4\)
(d) \(8\)
Solution
\[ (x+y)^3=x^3+3x^2y+3xy^2+y^3 \]
\[ (x-y)^3=x^3-3x^2y+3xy^2-y^3 \]
\[ (x+y)^3-(x-y)^3 =6x^2y+2y^3 \]
\[ 6y(x^2-y^2)=6x^2y-6y^3 \]
\[ (6x^2y+2y^3)-(6x^2y-6y^3) \]
\[ =8y^3 \]
Therefore,
\[ k=8 \]
\[ \boxed{(d)\ 8} \]