Is it possible to find three quadratic functions $f(x),g(x)$ and $h(x)$ such that $f(g(h(x)))$ has $-6,-5,-4,-2,1,3,4,5$ as its roots?
I understand that the composition of three quadratic functions has $2^{3}=8$ roots, so this satisfies the condition. I also know that $f(g(h(x)))=k(x-1)(x-3)(x-4)(x-5)(x+2)(x+4)(x+5)(x+6)$ for some $k\in \mathbb{R}$. I have no idea what's next. It seems too complicated to make it in the form of composition of three $ax^2+bx+c$. What are some properties of the composition of quadratic functions I could use? Thanks.
Let $h(x)=(x+2)(x-1)$, so $h(-6)=h(5)=28$, $h(-5)=h(4)=18$, $h(-4)=h(3)=10$, and $h(-2)=h(1)=0$. Let $g(x)=x(x-28)$, so $g(28)=g(0)=0$, $g(18)=g(10)=-180$. Then let $f(x)=x(x+180)$.