If x+y+z=0, then what is the value of frac{x^ | vedclass

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(A) Given that $x+y+z=0.$We need to find the value of the expression $E = \frac{x^{2}}{3yz} + \frac{y^{2}}{3xz} + \frac{z^{2}}{3xy}.$Taking the least

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$0$

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(A) Given that $x+y+z=0.$We need to find the value of the expression $E = \frac{x^{2}}{3yz} + \frac{y^{2}}{3xz} + \frac{z^{2}}{3xy}.$Taking the least common multiple $(LCM)$ of the denominators $3yz, 3xz,$ and $3xy,$ which is $3xyz,$$E = \frac{x^{2}(x) + y^{2}(y) + z^{2}(z)}{3xyz} = \frac{x^{3} + y^{3} + z^{3}}{3xyz}.$We know the algebraic identity that if $x+y+z=0,$ then $x^{3} + y^{3} + z^{3} = 3xyz.$Substituting this into our expression:$E = \frac{3xyz}{3xyz} = 1.$Note: Based on the provided options and the structure of the expression,the result is $1$. If the expression was $\frac{x^3}{3yz} + \dots$ or similar,the result would vary. Given the standard form of this identity,the value is $1$.

If $x+y+z=0,$ then what is the value of $\frac{x^{2}}{3yz} + \frac{y^{2}}{3xz} + \frac{z^{2}}{3xy}?$

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