SOLUTION: If x + y + z = 6 x² + y² + z² = 38 x³ + y³ + z³ = 144 Find x, y and z

x + y + z = 6,        (1)

 = 38,     (2)

 = 144.    (3)

        Preliminary notice. Since the problem is a bit higher than the traditional school math, 
        the solution is, correspondingly, a bit higher (but still understandable). 

We are given the numerical values for the expressions x+y+z,    and  . 

Using it, we can explicitly express and calculate values  xy+yz+zx  and  xyz.

Next, knowing the numerical values of  x+y+z,  xy+yz+zx  and  xyz, we can consider the polynomial of the variable "u"

P(u) = .   (4)

This polynomial is nothing else as 

P(u) = (u-x)*(u-y)*(u-z).     (5)

In other words, the polynomial (4) is factorable into (5).

Now, instead of solving the system (1), (2), and (3), we can solve the polynomial equation 

P(u) = 0      (6)

for u. If we solve it (and when we solve it), its roots u = x, u = y and u = z will be the solution to the system (1), (2), (3).

Why this way is better than solving (1) directly?
Well, for example, you can (try to) solve the polynomial equation (4) graphically. 
Or apply other methods specific for polynomial equations. 
You will see it later.

Now I will implement this methodology.

    It is easy. From (1), we have

        x + y + z = 6.

    Square both sides.  You will get

        x^2 + y^2 + z^2 + 2xy + 2xz + 2yz = 36.

    Replace here x^2 + y^2 + z^2 by 38, based on (2).  You will get

        2xy + 2xz + 2yz + 38 = 36,

        xy + xz + yz = -1.
    

    Thus part 2 is complete.

We will do it step by step:

1.   = .


2.    = 
   = x(xy+xz) + y(xy+yz) + z(xz+yz) = 
   = x(xy+xz+yz-yz) + y(xy+yz+xz-xz) + z(xz+yz+xy-xy)
   = x(xy+xz+yz) + y(xy+xz+yz) + z(xy+xz+yz) - x(yz) - y(xz) - z(xy)
   = (x+y+z)(xy+xz+yz) - 3xyz.


3.  Therefore, 

      = ,   or

      = ,   or

      = ,   or

     xyz = . 


4.   Thus

         xyz =  = -30.


     Part 3 is complete.

So, our polynomial (4) is

    P(u) = ,

and we need solve this polynomial equation 

     = .

First, let's do it graphically.

Figure. Plot P(u) =

Do you see the roots from the plot?  But of course, they are u= -2, u=3 and u=5.

And you can check it manually substituting these values into the polynomial.

Or you can apply the rational roots theorem.

According to this theorem, all the roots are among the integer divisors of the number 30, 
and you have only finite number of options to check. It is your other method to find the roots.