If \(x+y+z=0\), then what is \(x(y+z)^2+y(z+x)^2+z(x+y)^2\) equal to?
3xyz
Since \(x+y+z=0\), we have \(y+z=-x,\ z+x=-y,\ x+y=-z\). So the expression becomes \(x(-x)^2+y(-y)^2+z(-z)^2=x^3+y^3+z^3\). Using the identity that when \(x+y+z=0\), \(x^3+y^3+z^3=3xyz\), the expression equals \(3xyz\). The correct option is (c).
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