Question 59 (a)

What should be added to $x^{3} + 3 x^{2} y + 3 x y^{2} + y^{3}$ to get $x^{3} + y^{3} ?$

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Solution

In order to find the solution subtract $x^{3} + 3 x^{2} y + 3 x y^{2} + y^{3}$ from $x^{3} + y^{3} .$
Required expression is
$x^{3} + y^{3} - (x^{3} + 3 x^{2} y + 3 x y^{2} + y^{3}) = x^{3} + y^{3} - x^{3} - 3 x^{2} y - 3 x y^{2} - y^{3}$
On combining the like terms.
$= x^{3} - x^{3} + y^{3} - y^{3} - 3 x^{2} y - 3 x y^{2} = - 3 x^{2} y - 3 x y^{2}$
So, if we add $- 3 x^{2} y - 3 x y^{2}$ in $x^{3} + 3 x^{2} y + 3 x y^{2} + y^{3}$ , we get $x^{3} + y^{3} .$

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The expansion of ${(x + y)}^{3}$ is

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