(distinct=different ; maybe this is a "Gallicism", I don't know...)

What is essential is that we know what the roots of P are (for we need to know what there sum is), together with their multiplicity (here, they are simple). On one hand, we know there are at most m of them (by the degree of P). On the other hand, we have m different ones at our disposal, namely

. As a consequence, the roots

__are__ .

There could have double roots, provided we would have known which ones they were (and count them twice in the sum of the roots).

If the degree of

was

, then there would be one root missing, and the sum of the roots would be undeterminated by the only knowledge of

.