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Descartes’ Method for Constructing Roots of Polynomials with ‘Simple’ Curves - Roots of Polynomials in Today’s Algebra Curriculum

Gary Rubinstein (Stuyvesant High School)

In beginning 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. 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. The graphical method is also limited since it only provides an approximate answer in many cases.

A first degree equation in one variable can be solved graphically by first converting it into a system of equations. For example, \(2x+1=5\) can be solved graphically by finding the \(x\)-coordinate of the intersection point of the lines defined by \(y=2x+1\) and \(y=5.\) It could also be solved using the lines defined by many other pairs of equations; for example, \(y=2x\) and \(y=4,\) or \(y=2x-4\) and \(y=0\) (the \(x\)-axis). As this method is more complicated than solving the first degree equation algebraically, it is not recommended for first degree equations.

Quadratic equations are sometimes solved graphically in Algebra I or Algebra II classes with the aid of a graphing calculator. The equation \(x^2-5x+6=0\) can be solved by finding the intersection points of the parabola defined by \(y=x^2-5x+6\) and the horizontal line defined by \(y=0\) (the \(x\)-axis). Another way is to find the intersection of the parabola defined by \(y=x^2\) and the line defined by \(y=5x-6.\) Without graphing technology, this method would be completely impractical since creating the parabola defined by \(y=x^2-5x+6\) would require finding the \(x\)-intercepts first. But once the \(x\)-intercepts are found, the original equation \(x^2-5x+6=0\) would already be solved so there would be no reason to find the solution graphically.

Higher degree equations like \(y=x^4-4x^3-16x^2+20=0\) can be solved graphically with a graphing calculator by finding the \(x\)-intercepts of \(y=x^4-4x^3-16x^2+20.\) But without graphing technology, making the graph would require finding the \(x\)-intercepts by solving the original equation using algebra or numerical methods. Without using graphing technology to ‘cheat’, it seems that solving equations of degree 2 or higher graphically is a hopeless and unnecessary endeavor.

Gary Rubinstein (Stuyvesant High School), "Descartes’ Method for Constructing Roots of Polynomials with ‘Simple’ Curves - Roots of Polynomials in Today’s Algebra Curriculum," Convergence (March 2016)