= Solution
Write $q=i\omega$. A constant-coefficient <linear ordinary differential equation> has <transfer function> $1/P(q)$, where $P(q)=\sum_{n=1}^c a_nq^n$. The specified simple <poles> correspond to $q=0,-2,1$. A minimal-order choice is therefore
$$
P(q)=q(q+2)(q-1)=q^3+q^2-2q.
$$
Thus one possible system is
$$
\boxed{x'''+x''-2x'=F,\qquad c=3,\quad(a_1,a_2,a_3)=(-2,1,1),}
$$
up to a common nonzero multiplicative constant. The root $q=1$ permits an exponentially growing homogeneous solution; the pole data therefore describe an unstable system. This does not prevent the requested algebraic reconstruction of its <transfer function>.
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