If $$x + y + z = 10, xy + yz + zx =25$$ and $$xyz = 100$$, then what is the value of $$(x^3 + y^3 + z^3)?$$

Given that$$x +y+ z = 10 , xy+yz+zx = 25   and    xyz = 100 $$

formula, $$x^3+y^3+z^3 - 3xyz = (x+y+z)(x^2+y^2+z^2-xy-yz-zx)$$ put the value

$$\Rightarrow x^3+y^3+z^3 - 3\times 100 = 10 [ x^2 +y^2 + z^2 - (xy+yz+zx)$$

$$\Rightarrow x^3+y^3 +z^3 -300 = 10 [x^2+y^2+z^2 - 25] $$..........equestion (1)

Further ,

$$x+y+z = 10 $$ 

$$\Rightarrow (x+y+z)^2 = 10^2 $$ (squaring both side)

$$\Rightarrow x^2+y^2+z^2 + 2 (xy+yz+zx) = 100 $$

$$\Rightarrow x^2+y^2+z^2 + 2\times 25 = 100 $$

$$\Rightarrow x^2+y^2+z^2 = 100-50 $$ 

$$\Rightarrow x^2+y^2 + z^2 = 50 $$ 

put the value $$x^2 +y^2 +z^2 = 50$$ in equestion (1) 

$$\Rightarrow x^3 +y^3 +z^3 - 300 = 10 [50-25]$$ 

$$\Rightarrow x^3 +y^3 +z^3 = 10 \times 25 \times 300 $$ 

$$\Rightarrow x^3 +y^3 +z^3 = 550 Ans $$