Vieta's formulas
In algebra, Vieta's formulas are a set of results that relate the coefficients of a polynomial to its roots. In particular, it states that the elementary symmetric polynomials of its roots can be easily expressed as a ratio between two of the polynomial's coefficients.
It is among the most ubiquitous results to circumvent finding a polynomial's roots in competition math and sees widespread usage in many mathematics contests.
Statement
Let be any polynomial with complex coefficients with roots
, and let
be the
elementary symmetric polynomial of the roots.
Vieta’s formulas then state that
This can be compactly summarized as
for some
such that
.
Proof
Let all terms be defined as above. By the factor theorem, . We will then prove Vieta’s formulas by expanding this polynomial and comparing the resulting coefficients with the original polynomial’s coefficients.
When expanding this factorization of , each term is generated by a series of
choices of whether to include
or the negative root
from every factor
. Consider all the expanded terms of the polynomial with degree
; they are formed by multiplying a choice of
negative roots, making the remaining
choices in the product
, and finally multiplying by the constant
.
Note that adding together every multiplied choice of negative roots yields
. Thus, when we expand
, the coefficient of
is equal to
. However, we defined the coefficient of
to be
. Thus,
, or
, which completes the proof.
Problems
Here are some problems with solutions that utilize Vieta's formulas:)