AMC 12 Mock Exam 5 Problem 24

A cubic with real coefficients has a real root, which must be $1$ or $-1$. So the polynomial is $(x-\varepsilon)Q(x)$ with $\varepsilon=\pm1$ and $Q(x)=x^2+px+q$ monic with integer coefficients (by polynomial division by $x-\varepsilon$), whose two roots lie on the unit circle.

Quadratics $Q$. If $Q$ has nonreal roots $w,\bar w$, then $q=|w|^2=1$ and $p=-2\operatorname{Re}w$ with $|p|<2$: $Q\in\{x^2+1,\ x^2+x+1,\ x^2-x+1\}$. If $Q$ has real roots, they are $\pm1$: $Q\in\{(x-1)^2,\ (x+1)^2,\ x^2-1\}$. That is $6$ quadratics.

Products. The $2\cdot6=12$ products $(x\mp1)Q(x)$ give each cubic once, except that $(x-1)(x^2-1)=(x+1)(x-1)^2$ and $(x+1)(x^2-1)=(x-1)(x+1)^2$ are each produced twice. The distinct cubics number $12-2=\boxed{10}$: $$(x-1)^3,\ (x+1)^3,\ (x-1)^2(x+1),\ (x+1)^2(x-1),\ (x\pm1)(x^2+1),\ (x\pm1)(x^2+x+1),\ (x\pm1)(x^2-x+1).$$