In the AOPS vol 2 problem solving book, it states that you can find the sum of the reciprocals of a polynomial by flipping the coefficients(first -> last, last -> first etc). The book summarized the fact that you can "transform polynomials" to get a desired sum of roots (just like the example I mentioned above). However, I'm confused on how to do it for more complicated sums, so could anyone explain the general method of transforming polynomials.
Here is an example question to demonstrate:
If the roots of polynomial $x^4−3x^3−27x^2−13x+42$ are $r_1$, $r_2$, $r_3$, $r_4$, find $(1/(r_1 + 1))$+$(1/(r_2 + 1))$+$(1/(r_3 + 1))$+$(1/(r_4 + 1))$.