In Chapter II of Dirac's book Principles of Quantum Mechanics, Dirac explains that in general it is very difficult to know whether, for a given real linear operator, that any eigenvalues/eigenvectors exist and (if they do) how to find them.
He then goes on to state that a special tractable case can be found in the event that a real linear operator, $\xi$ can be expressed as the algebraic expression:
$\phi$($\xi$) $\equiv$ $\xi$n + a1 $\xi$n-1 + a2 $\xi$n-2 + ... + an = 0
Where the ai are all numbers. He then factorises this as:
$\phi$($\xi$) $\equiv$ ($\xi$- c1)($\xi$- c2)($\xi$- c3)...($\xi$- cn).
Where the ci are also numbers.
I'm unsure how this factorisation has been achieved. Since Dirac didn't explicitly state the difference between $\phi$($\xi$) and plain $\xi$, then I imagine that $\xi$ represents a given dynamical variable and $\phi$($\xi$) the corresponding operator. This may be incorrect however. This is the only step in the logic that I can't work out, I understand his subsequent arguments but this step is eluding me.