Albert Girard (1590-1633), a flemish mathematician, published in 1629, in
Amsterdam, a book called Invention nouvelle en l'algèbre, in which
clear relations between roots and coefficients of polynomials were stated
for the first time. Let where .
Girard had the brilliant idea of postulating that such a polynomial had
to have n roots; whenever a polynomial had less than n real roots, one could
extend the set of the roots to have order n by adjoining complex
roots. Under these conditions, one could write
where is the root. Hence
the derivative is given by
Notice that for each we have
Now, provided , we have
Now let . This means
which gives infinite relations between coefficients and roots. Take for example
Although it is possible to derive infinitely many such relations, to isolate one single root one needs at least a difference between some powers of the roots. In the case of the quadratic we have
hence we can isolate the roots and solve the equation.