Relation between Roots and Coefficients

An algebraic equation in one variable $x$ is an equation of the form $f(x)=0$, where $f(x)=a_0x^n+a_1x^{n-1}+\cdots+a_{n-1}x+a_n$ is a polynomial of degree $n$ (so $a_0\neq0$). The degree of the equation is $n$, the highest power of $x$ occurring in it. When $a_0=1$ the equation is called a monic equation. Every equation of degree $n$ has exactly $n$ roots (real or complex, counted with multiplicity) -- a fact we take as known from the Fundamental Theorem of Algebra.

This chapter builds a toolkit for working with an equation through its roots, often without ever solving it explicitly. Two ideas run through everything that follows. First, the coefficients of an equation are not arbitrary numbers: they are built out of the symmetric functions of the roots -- their sum, the sum of their pairwise products, and so on -- and this two-way dictionary between coefficients and root-combinations is the single most useful fact in the whole chapter. Second, when the roots of an equation are known to satisfy some extra condition -- they might be in progression, some might repeat, some might be complex or irrational -- that extra condition combines with the coefficient relations to pin the roots down completely, or at least reduce the work of finding them.