Find value of $x^{5} + y^{5} + z^{5}$ given the values of $x + y + z$, $x^{2} + y^{2} + z^{2}$ and $x^{3} + y^{3} + z^{3}$ If$$x+y+z=1$$
$$x^2+y^2+z^2=2$$
$$x^3+y^3+z^3=3$$
Then find the value of $$x^5+y^5+z^5$$
Is there any simple way to solve this problem ? I have tried all my tricks tried to multiply two equations , substitute $z=1-x-y$ , but things got messy nothing seems to work out .

HINT
Multiply the third relation by the second so that you obtain
\begin{align*}
(x^{2} + y^{2} + z^{2})(x^{3} + y^{3} + z^{3}) & = x^{5} + y^{5} + z^{5} + x^{2}(y^{3} + z^{3}) + y^{2}(x^{3} + z^{3}) + z^{2}(x^{3} + y^{3})
\end{align*}
Now we can rearrange the last expression as
\begin{align*}
x^{2}y^{2}(x + y) + x^{2}z^{2}(x + z) + y^{2}z^{2}(y + z) & = x^{2}y^{2} + x^{2}z^{2} + y^{2}z^{2} - x^{2}y^{2}z - x^{2}yz^{2} - xy^{2}z^{2}\\\\
& = x^{2}y^{2} + x^{2}z^{2} + y^{2}z^{2} - xyz(xy + xz + yz)
\end{align*}
Hence the problem has been reduced to study the last relation.
Before keep going, notice that
\begin{align*}
x^{2}y^{2} + x^{2}z^{2} + y^{2}z^{2} & = (xy + xz + yz)^{2} - 2xyz(x + y + z)\\\\
& = (xy + xz + yz)^{2} -2xyz
\end{align*}
In order to find the remaining expression, let us observe that
\begin{align*}
(x + y + z)^{2} = x^{2} + y^{2} + z^{2} + 2(xy + xz + yz) \Rightarrow xy + xz + yz = -\frac{1}{2}
\end{align*}
At last but not least, we have to find the value of $xyz$.
Can you take it from here?
