Descartes’ Method For Constructing Roots Of Polynomials With ‘Simple’ Curves - Introduction

Gary Rubinstein (Stuyvesant High School)

Descartes' 1637 Geometry presented symbolically and graphically via interactive applets ... coming soon!

In elementary algebra classes, soon after students learn to graph solution sets of linear relations, they learn that it is then possible to ‘graphically solve’ a system of two equations in two unknowns by carefully graphing both solution sets and locating the coordinates of the intersection point. Thus, in schools today, students are taught that systems of linear equations can be solved graphically by finding the intersection of two lines.  Later, students learn that these same systems of equations can be solved algebraically using substitution or elimination.  The algebraic method is preferred since it is both quicker and easier.  It is also more accurate since it does not rely on eyeballing the coordinates of the intersection point.

For higher degree equations, graphical solutions are possible with graphing calculators, which certainly feels like cheating.  Without graphing technology, graphically solving a higher degree equation does not seem feasible since producing a graph of a higher degree curve generally requires solving a problem at least as hard as the problem you are trying to solve algebraically.

So two aspects of linking algebra and geometry via coordinate geometry have been established: 

  1. It is possible to solve algebraic equations by intersecting appropriate curves, and
  2. It is possible to locate the intersections of curves with algebra. 

But soon after the algebraic method is taught, ... coming soon!

Figure 1. Constructing a line segment of length \(\sqrt{n}.\) Instructions: In the sketch, drag the point on the slider to change the value of \(n.\)

Along with ‘squaring the circle’ and ‘trisecting the angle,’ the third great problem of antiquity was ‘doubling the cube.’  This problem required constructing – using only compass and straightedge – a line segment so that a cube constructed on that segment would have twice the volume of a cube constructed on a given segment.  In modern terms, this requires constructing a segment of length equal to the length of the edge of the original cube multiplied by \(\sqrt[3]{2}.\) ... Coming soon!

Figure 6: Constructing the Cartesian Parabola

Instructions: Move the sliders to adjust \(a, b,\) and \(n.\) Move the vertex of the parabola up and down to see how the two branches are formed.

The equation for the two-branched Cartesian parabola, which will be derived later (see Deriving the Formula for the Cartesian Parabola ... coming soon!), is the rational equation \[y=\frac{x^{2}}{n}-\frac{bx}{n}-a+\frac{ab}{x}.\] Notice that this rational function has a vertical asymptote at \(x=0\) as well as a parabolic asymptote. Because of these features, the curve is a favorite with teachers of pre-calculus classes. What makes this curve ‘simple’ is that it is generated by only a line and a parabola, as opposed to, say, a graph of a cubic polynomial.