Solution of simultaneous equations using properties of symmetric polynomials

Question

Solve the following system of equations.

$$ x + y + z = 5 $$ $$ \frac{1}{xy}+\frac{1}{yz}+\frac{1}{zx} = 5 $$ $$ x^3 + y^3 + z^3 = 53 $$

How would I use properties of symmetric polynomials (e) to solve this?

Answer 1

.From $\frac{1}{yz} + \frac{1}{xz} + \frac{1}{xy} = 5$, we get: $$ \frac{x+y+z}{zyx} = 5 \implies zyx = 1 $$

Now, $$ x^3+y^3+z^3 - 3xyz= (x+y+z)(x^2+y^2+z^2 - xy-yz-zx) \\= (x+y+z)((x+y+z)^2-3(xy+yz+zx)) $$ Solving using values, this gives $xy+yz+zx = 5$.

Now, note that: $$ (t-x)(t-y)(t-z) = t^3 - (xy+yz+zx)t^2 + (x+y+z)t - xyz \\= t^3 - 5t^2 + 5t-1 $$ Hence, if we are able to find the roots of this equation, we can find the values of $x,y,z$.

By inspection, $t=1$ works, hence $t-1$ ia a factor.

After the division ,we are left with $t^2-4t+1$, which gives $t = 2 \pm \sqrt{3}$.

Thus, $x=1, y,z = 2 \pm \sqrt{3}$, and since the equation is symmetric, any two of these can be switched.