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 all 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 $-1)^$ (Error compiling LaTeX. Unknown error_msg)(-1)^j s_j
P(x)
x_{n-j}
(-1)^j a_n s_j
x^{n-j}
a_{n-j}
(-1)^j a_n s_j = a_{n-j}
s_j = (-1)^j a_{n-j}/a_n
\Box$
Problems
Here are some problems with solutions that utilize Vieta's formulas.