= . 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 and ax by = 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.
Abel ( - ) - - 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 sides are constructible whereas a regular heptagon and a regular polygon of 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