Mathematics · Ch 3 — Theory of Equations

Introduction

3.1

Introduction

Solving polynomial equations is one of the oldest pursuits in mathematics — Sumerian and Babylonian scribes were already working with them around 2000 BCE, and mathematicians across Egypt, Greece, India, Arabia and China all attempted them in their own notations, long before a general symbolic algebra existed. Brahmagupta was the first to solve quadratic equations allowing negative numbers. Later, Ruffini argued (in a proof many found hard to follow) that no algebraic formula could solve a general degree-5 equation, and in 1823 the Norwegian mathematician Abel proved this rigorously: there is no radical formula — built only from $+,-,\times,\div$ and root-extraction — that solves every degree-5-or-higher polynomial equation. This is exactly why this chapter's toolkit (Vieta's formulae, spotting extra structure, Descartes' Rule, ...) matters: for degree $\ge5$ , these clever techniques are often the only way in, since there is no universal formula to fall back on.

Why bother with a whole chapter of technique? Two motivating examples:

Note

The box problem. A company wants to pack its product into a box whose base has length $6$ units more than its breadth, and whose height is the average of the length and the breadth. If the breadth is $x$ , the length is $x+6$ and the height is $x+3$ , so the volume is $x(x+6)(x+3)$ . If the volume must equal $2618$ cubic units, we need $x^3+9x^2+18x=2618$ — finding a valid $x$ means solving a cubic equation.

The circle-and-line problem. A circle $x^2+y^2=r^2$ and a line $ax+by+c=0$ meet at the points that satisfy both equations simultaneously. Substituting one into the other collapses this geometry question into a single polynomial equation in one variable — so how many roots that equation can have directly tells us how many points the line and circle can share.

Some classical construction problems — trisecting an arbitrary angle, squaring a circle with compass and straightedge (Ramanujan gave a famous approximate construction), doubling a cube — turn out to be genuinely impossible, and this was only settled by converting them into questions about polynomial equations. Mathematics, used this way, is a powerful tool for proving impossibility, not just for computing answers.

What this chapter covers. Vieta's Formula for polynomial equations of degree $2$ , $3$ and $n>3$ ; forming a polynomial equation from given roots (or from a transformation of a known equation's roots); the Fundamental Theorem of Algebra; what the nature of the coefficients (real / rational / integer) forces about the nature of the roots (the Complex Conjugate Root Theorem and its surd-root cousin); solving higher-degree equations using extra information (progressions, partly-factored quartics, reciprocal equations); the Rational Root Theorem for when no extra information is given; and Descartes' Rule of Signs for bounding how many positive, negative and non-real roots an equation can have — all without necessarily solving the equation.

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