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, . When we expand this polynomial, each term is generated by the
choices of whether to include
or
from any factor
. We will then prove Vieta’s formulas by expanding this polynomial and comparing the resulting coefficients with the original polynomial’s coefficients.
Consider all the expanded terms of with degree
; they are formed by choosing
of the negative roots, making the remaining
choices
, and finally multiplied by the constant
. We note that when we multiply
of the negative roots, we get
.
So in mathematical terms, 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 that test knowledge of Vieta's formulas.