I am trying to answer the following exercise:
I have managed to almost answer it, I noticed that I need to use the answer from another exercise:
I just made some calculations a little bit wrong. I went for the solutions of the book and found:
Which is akin to what I used in my answer. And there is a problem with both this answer and my "answer" with mistaken calculations: Using the previous exercise, we know that $\frac{-31}{36}=\frac{-b}{a}$ and $\frac{-449}{216}=\frac{c}{a}$, but why do we have two different $a$'s? I thought that there could be two polynomials with different coefficients and same zeroes, but at least here, it is not the case:
So how do I pick the right $a$?
$$m+n=5/6\\mn=-1/2$$ If the roots are $m-n^2$ and $n-m^2$ then the sum of roots is $$m-n^2+n-m^2=(m+n)-(m+n)^2+2mn=?$$ And product of roots. $$(m-n^2)(n-m^2)=mn-n^3-m^3+m^2n^2=mn + m^2n^2-(m+n)(m^2+n^2-mn)\\=mn+m^2n^2-(m+n)((m+n^2)-3mn)$$
Put the values of $m+n$ and $mn$ to get sum and product of roots. This is the answer and there is no right $a$. You need to have the correct ratio between $a,b,c$. Their absolute values don't really matter.