An algebra problem by Hana Wehbi

Algebra Level 3

If x , y x,y and z z are non-zero real numbers satisfying x + y + z = 0 x+y+z= 0 , what is the value of 11 ( x 2 y z + y 2 x z + z 2 x y ) 2 11 \left(\dfrac{x^2}{yz} + \dfrac{y^2}{xz} + \dfrac{z^2}{xy} \right)^2 ?

39 99 11 121

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Hana Wehbi by Hana Wehbi , 51, USA

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1 solution

Hana Wehbi
May 23, 2016

x + y + z = 0 ⟹ x 3 + y 3 + z 3 = 3 x y z x+y+z=0 \implies x^{3}+y^{3}+z^{3}=3xyz

Multiplying the numerators and denominators of the given expression by x x , y y and z z respectively,

( x 2 y z \large\frac{x^{2}}{yz} + y 2 x z \large\frac{y^{2}}{xz} + z 2 x y \large\frac{z^{2}}{xy} ) = x 3 x y z \frac{x^{3}}{xyz} + y 3 x y z \frac{y^{3}}{xyz} + z 3 x y z \frac{z^{3}}{xyz} = x 3 + y 3 + z 3 x y z \frac{x^{3}+y^{3}+z^{3}}{xyz} = 3 x y z x y z \frac{3xyz}{xyz} = 3

Keeping in mind x x , y y and z z are not equal to 0 0 .

Thus, our answer is 11 ∗ 3 2 = 99 11*3^{2}=99

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