How to define $z$ with $x$ and $y$ using implicit function theorem?

Question

Let $c$ be a nonzero constant. Consider the fuction

$$f(x,y,z) =x^3yz+xyz^3=c.$$

How to define $z$ as a function of $x$ and $y$ ?

How to use the implicit function theorem to solve this kind of questions?

I tried to solve it by manipulating variable $z$. However, such question cannot be solved in this way. I can solve by implicit with functions like $f(x,y)=x^2+y+1$ or $f(x)=2x^2+4$. As soon as basic move failed to separate the desired variable, I can't solve it anymore. Separating $xyz$? or using differentiation ? I tried with partial derivative, but it's way to complicate.

I really would like to know if there is any technique to solve this.

Answer 1

The implicit function theorem won't help you find an expression for $z$; it just gives local existence of a corresponding function $z(x,y)$. In this case, you have a cubic equation in $z$, so you can solve it using algebraic techniques, e.g. the generally very ugly Cubic formula.