I have recently discovered Vieta jumping as a problem-solving technique. In order to teach myself about it, I have located most (all of?) the standard references, both here on MSE and "out there" (via web searches).

However, all the examples I've found are of a form where the equation to be subjected to root-flipping is a monic [quadratic] polynomial, e.g., $$x^2+rx+s = 0.$$ This, of course, is quite useful, because then Vieta's root equation $$ x_2 = -\frac{B}{A} - x_1 = -r-x_1$$ implies that $x_2$ must be an integer when $r$ and $x_1$ are integers. On the other hand, when $A > 1$ we have instead $$ x_2 = -\frac{B}{A} - x_1 = -\frac{r}{t}-x_1$$ for some integer $t > 1$, so one can't immediately conclude that $x_2$ is an integer. (I am currently trying to solve a problem just like this.)

Does anyone have an example of a problem solved by Vieta jumping [at least partly], where the root-flipping equation is not monic?

Thanks, Kieren.


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