The name root linear coefficient theorem is used here for the following consequence of Vieta's formulas. For a polynomial 
of degree 
with 
and roots 
, ..., 
counted with multiplicity,
 |
In particular, when 
,
the sum of the reciprocals of the roots
is the negative of the linear coefficient.
The identity follows by comparing linear coefficients in the factorization
 |
The corresponding assertion does not hold for an arbitrary Maclaurin series. For example, the exponential function 
has constant and linear coefficients equal to 1 but
has no roots. Applications to infinite
products require a justified product representation and coefficient
comparison.
