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Find the value of x^{3} + y^{3} − 12xy + 64, when x + y =−4

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#### Solution

The given expression is

`x^3 +y^3 - 12xy +64`

It is given that

`x+y = -4`

`⇒ x+y+4 = 0`

The given expression can be written in the form

`x^3+y^3 -12xy +64 = x^3 +y^3+ 64 -12xy`

` = (x)^3 + (y)^3 + (4)^3 - 3.(x).(y).(4)`

Recall the formula

`a^3+b^3 +c^3 -3abc = (a+b+c)(a^2+b^2 +c^2 - ab -bc - ca)`

Using the above formula, we have

`x^3 +y^3 -12xy +64`

`= (x+y+4){(x)^2 + (y)^2 + (4)^2 - (x).(y) - (y).(4) -(4).(x)}`

` = (x+y+4)(x^2 +y^2 +16 - xy -4y -4x)`

` = 0.(x^2 +y^2 +16 - xy -4y-4x)`

` = 0`

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