Introduction

Theory of Equations

“It seems that if one is working from the point of 

view of getting beauty in one’s equation, 

and if one has really a sound insight, one is on a sure line of progress.”

- Paul Dirac

Introduction

One of the oldest problems in mathematics is solving algebraic equations, in particular, finding the roots of polynomial equations. Starting from Sumerian and Babylonians around 2000 BC (BCE), mathematicians and philosophers of Egypt, Greece, India, Arabia, China, and almost all parts of the world attempted to solve polynomial equations.

The ancient mathematicians stated the problems and their solutions entirely in terms of words. They attempted particular problems and there was no generality. Brahmagupta was the first to solve quadratic equations involving negative numbers. Euclid, Diophantus, Brahmagupta, Omar Khayyam, Fibonacci, Descartes, and Ruffini were a few among the mathematicians who worked on polynomial equations. Ruffini claimed that there was no algebraic formula to find the solutions to fifth degree equations by giving a lengthy argument which was difficult to follow; finally in 1823, Norwegian mathematician Abel proved it.

Suppose that a manufacturing company wants to pack its product into rectangular boxes. It plans to construct the boxes so that the length of the base is six units more than the breadth, and the height of the box is to be the average of the length and the breadth of the base. The company wants to know all possible measurements of the sides of the box when the volume is fixed.

If we let the breadth of the base as x , then the length is x + 6 and its height is x + 3 . Hence the volume of the box is x(x + 3)(x + 6) . Suppose the volume is 2618 cubic units, then we must have x³ + 9x² +18x = 2618 . If we are able to find an x satisfying the above equation, then we can construct a box of the required dimension.

We know a circle and a straight line cannot intersect at more than two points. But how can we prove this? Mathematical equations help us to prove such statements. The circle with centre at origin and radius r is represented by the equation x² + y² = r² , in the xy -plane. We further know that a line, in the same plane, is given by the equation ax + by + c = 0 . The points of intersection of the circle and the straight line are the points which satisfy both equations. In other words, the solutions of the simultaneous equations

x² + y² = r² and ax + by + c = 0

give the points of intersection. Solving the above system of equations, we can conclude whether they touch each other, intersect at two points or do not intersect each other.

There are some ancient problems on constructing geometrical objects using only a compass and a ruler (straight edge without units marking). For instance, a regular hexagon and a regular polygon of 17 sides are constructible whereas a regular heptagon and a regular polygon of 18 sides are not constructible. Using only a compass and a ruler certain geometrical constructions, particularly the following three, are not possible to construct: 

·               Trisecting an angle (dividing a given angle into three equal angles).

·               Squaring a circle (constructing a square with area of a given circle). [Srinivasa Ramanujan has given an approximate solution in his “Note Book”]

·               Doubling a cube (constructing a cube with twice the volume of a given cube).

These ancient problems are settled only after converting these geometrical problems into problems on polynomials; in fact these constructions are impossible. Mathematics is a very nice tool to prove impossibilities.

When solving a real life problem, mathematicians convert the problem into a mathematical problem, solve the mathematical problem using known mathematical techniques, and then convert the mathematical solution into a solution of the real life problem. Most of the real life problems, when converting into a mathematical problem, end up with a mathematical equation. While discussing the problems of deciding the dimension of a box, proving certain geometrical results and proving some constructions impossible, we end up with polynomial equations.

In this chapter we learn some theory about equations, particularly about polynomial equations, and their solutions; we study some properties of polynomial equations, formation of polynomial equations with given roots, the fundamental theorem of algebra, and to know about the number of positive and negative roots of a polynomial equation. Using these ideas we reach our goal of solving polynomial equations of certain types. We also learn to solve some non–polynomial equations using techniques developed for polynomial equations.

Learning Objectives

Upon completion of this chapter, the students will be able to

● form polynomial equations satisfying given conditions on roots.

● demonstrate the techniques to solve polynomial equations of higher degree.

● solve equations of higher degree when some roots are known to be complex or surd, irrational, and rational.

● find solutions to some non-polynomial equations using techniques developed for polynomial equations.

● identify and solve reciprocal equations.

● determine the number of positive and negative roots of a polynomial equation using Descartes Rule.