Summary

In this chapter we studied:

• Vieta's Formula for polynomial equations of degree $2$, $3$, and $n>3$ — the relations linking coefficients to sums/products of roots, used both to read off root-combinations from an equation and to construct new equations from known (or transformed) roots.
• The Fundamental Theorem of Algebra: a polynomial of degree $n\ge1$ has at least one root in $\mathbb C$ — and, combined with the "at most $n$ roots" fact, exactly $n$ roots counted with multiplicity.
• The Complex Conjugate Root Theorem: non-real (imaginary) roots of a real-coefficient polynomial always occur in conjugate pairs — together with its rational-coefficient cousin for surd roots.
• The Rational Root Theorem: for $a_nx^n+\cdots+a_0=0$ with integer coefficients ($a_n\ne0,a_0\ne0$), any rational root $p/q$ (lowest terms) must have $p\mid a_0$ and $q\mid a_n$.
• Methods for special-structure equations: only even powers present, partly-factored quartics, coefficients summing to zero, and reciprocal equations. …