help

If

\(x^2 + y^2 + z^2 = a^2\) and \(x + y + z = x^3 + y^3 + z^3 = a\), then write \(xyz\) expicitly in terms of \(a\).

1.

\(\begin{array}{|rcll|} \hline (x+y+z)^2 &=& x^2 + y^2 + z^2+2(xy+xz+yz) \quad | \quad x + y + z = a,\ x^2 + y^2 + z^2 = a^2 \\ a^2 &=& a^2+2(xy+xz+yz) \\ 2(xy+xz+yz) &=& 0 \\ \mathbf{xy+xz+yz} &=& \mathbf{0} \\ \hline \end{array} \)

2.

Formula: \(x^3+y^3+z^3 = 3xyz+(x+y+z)\left(x^2+y^2+z^2-(xy+xz+yz)\right)\)

\(\begin{array}{|rcll|} \hline \underbrace{x^3+y^3+z^3}_{=a} &=& 3xyz+\underbrace{(x+y+z)}_{=a}\left(\underbrace{x^2+y^2+z^2}_{=a^2}-\underbrace{(xy+xz+yz)}_{=0}\right) \\ a &=& 3xyz+a ( a^2-0 ) \\ a &=& 3xyz+a^3 \\ 3xyz &=& a-a^3 \\ \mathbf{xyz} &=& \mathbf{ \dfrac{1}{3}(a-a^3) } \\ \hline \end{array} \)