TITLE: How do the roots of a polynomial change when another polynomial is added?
QUESTION [5 upvotes]: I need to obtain an analytical solution to an equation of the following form:
$$
(x-a)(x-b)(x-c)=d(x-e)(x-f),
$$
where $a$, $b$, $c$, $d$, $e$, and $f$ are known numbers and $x$ is the variable.
Of course, the equation can be reduced to a "simple" equation for the roots of a 3rd-order polynomial, and its solution is provided by the Cardano formula (and other similar methods), but the result is too complex...
So the question arises: does any simpler method exist that makes use of the known roots of the "constituent polynomials" (left-hand and right-hand side of my equation)?
P.S. It is possible to try the Cardano method with the polynomial coefficients expressed through the roots $a$...$c$,$e$,$f$ and then to simplify the resulting gargantuan formulae using a CAS (computer algebra system) hoping that the CAS will cope with it... But I am not so sure that the CAS is powerful enough in the dark art of formulae simplification.

REPLY [9 votes]: As you point out, this is no harder than solving a general cubic. It turns out that it is also no easier.
For any choice of the $6$ parameters $a,b,c,d,e,f$ your desired equation can be multiplied out to a cubic $x^3+Ax^2+Bx+C$ which can be solved by Cardano's formula. You hope for a nicer solution perhaps based on the form. However one might expect that with $6$ parameters one can easily get $A,B,C$ to be anything desired. This turns out to be the case.
Here is one way assuming all parameters constrained to be real. The complex case is as easy.
Pick $a=b=0$ and $c=-(A+d)$ where $d$ is $+1$ or $-1$ according as $C \ge 0$ or  $C \lt 0.$  Then $e$ and $f$ are the (real) roots $$\frac{Bd \pm \sqrt{B^2+4|C|}}{2}$$ of the quadratic equation.
$$x^2-\frac{B}{d}x-\frac{C}{d}.$$