Consider the trinomial \(T_{a,b}^{k,m}(\lambda)=\lambda^k-a\lambda^{k-m}-b\) with \(a,b\in\mathbb{R}\) and let \(n_d^-\), \(n_d^0\), \(n_d^+\) be the number of its zeros whose modulus is less than, equal to, or larger than \(d\), respectively. The following problems are solved:
(P1) Given \(a\), \(b\), \(m\), \(k\), \(d\), find formulas for \(n_d^-\), \(n_d^0\), \(n_d^+\).
(P2) Given \(k\), \(m\), \(d\), \(r\), find the couples \((a,b)\in\mathbb{R}^2\) such that \(n_d^0=r\).
(P3) Given \(k\), \(m\), \(d\), \(r\), find the couples \((a,b)\in\mathbb{R}^2\) such that \(n_d^-=r\) and \(n_d^+=k-r\).
The problems are simplified to \((k,m)\) being coprime and \(d=1\). A survey of the existing literature is given extended with new results. Solving the three problems results in extensive tables that list the conditions on the parameters that result in the different distributions of \(k=n_d^-+n_d^0+n_d^+\).