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 .

Here is yet another way, in case you are interested:
$$(x+y+z)^2=1\implies\sum x^2+2\sum xy=1\implies \sum xy=-\frac12$$
Therefore the polynomial equation whose roots are $x,y,z$ takes the form:
$$t^3-t^2-\frac12t+c=0$$
Summing this equation for each of the roots,
$$\sum x^3-\sum x^2-\frac12\sum x +3c=0$$
$$\implies3-2-\frac12(1)+3c=0\implies c =-\frac16$$
Multiplying the polynomial by $t$ and summing again,
$$\sum x^4-\sum x^3-\frac12\sum x^2-\frac16\sum x=0$$
$$\implies\sum x^4=3+\frac12(2)+\frac16(1)$$
$$\implies \sum x^4=\frac{25}{6}$$
Multiplying the polynomial by $t^2$ and summing again,
$$\sum x^5-\sum x^4-\frac12\sum x^3-\frac16\sum x^2=0$$
$$\implies\sum x^5=\frac{25}{6}+\frac12(3)+\frac16(2)=6$$
