Introduction

From One Equation to a General Theory

By this point you have solved linear equations, and quadratic equations and inequations, in real depth — including, from the previous chapter, the exact relationship between a quadratic's two roots and its coefficients $a,b,c$. But most equations that show up in real applications — engineering, economics, the sciences — have degree three or higher, and there is no simple formula for their roots the way the quadratic formula gives you two. The theory of equations is the collection of ideas that let us say precise, useful things about the roots of any polynomial equation of degree $n$, even when we cannot write those roots down explicitly.

What This Chapter Covers

This chapter extends the roots–coefficients relationship you already know for quadratics to polynomial equations of any degree $n$: the general relations between all $n$ roots and all $n$ coefficients, how to build a new equation whose roots are related to a known equation's roots in a prescribed way (for instance, doubled, or reciprocal), what happens to the complex roots of an equation with real coefficients (they always occur in conjugate pairs), and techniques for transforming one equation into another, including the special case of reciprocal equations.