If the expression `ax^(2)+by^(2)+cz^(2)+2ayz+2bzx+2cxy` can be resolved into two rational factors, prove that
`a^(3)+b^(3)+c^(3)=3abc`.

Options

A. abc
B. 3abc
C. 2abc
D. `-3abc`

Answer

B

Explanation

The given expression is `ax^(2) + by^(2) + cz^(2) + 2ayz + 2bzx + 2 cxy`
`=z^(2){a((x)/(z))^(2)+b((y)/(z))^(2)+c+2a((y)/(z))+2b((x)/(z))+2x((x)/(z))((y)/(z))}`
`=z^(2){aX^(2) + bY^(2)+ 2cXY+2bX+2aY+c}" "...(i)`
where `X = (x)/(z) and Y=(y)/(z)`.
The given expression can be resolved into rational factors if the expression within brackets in (i) is expressible into rational factors, the condition for which is
`abc + 2abc - a xx a^(2) - b xx b^(2) - c xx c^(2) = 0 rArr a^(3) + b^(3) + c^(3) = 3 abc`