In 1867 Eduard Lill gave the following surprising method for finding real roots of polynomial equations. Suppose the polynomial has the form [br][center][math]f\left(x\right)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0.[/math] [/center]1) First, we'll draw a polygonal path starting at the origin. If all the coefficients are positive, go [math]a_n[/math] units to the right, [math]a_{n-1}[/math] units up, [math]a_{n-2}[/math] units left, [math]a_{n-3}[/math] units down, and continue right, up, left, down, and so on. If any coefficient is negative, go in the opposite direction at that step. This path is blue in the applet.[br]2) Next, draw a line through the origin making an angle [math]\alpha[/math] with the [i]x[/i]-axis. This path is red in the applet.[br]3) Find where this red line intersects the first vertical blue line of the path constructed in step 1 (extended, if necessary). [br]4) Construct a perpendicular (also red) line to this red path at the point of intersection.[br]5) Find where this line crosses the second (blue) horizontal line (extended, if necessary).[br]6) Repeat as many times as required to generate the full red path.[br]7) If the ends of the red path and the blue path coincide for a given starting angle [math]\alpha[/math], then [math]-\tan\left(\alpha\right)[/math] is a real root of the polynomial.[br][br]In the applet below, change the sliders for [i]a[/i],[i] b[/i], [i]c[/i], and [i]d[/i] to generate different cubic equations. Then use the [math]\alpha[/math]-slider to change the starting angle of the red path. When the ends of the red and blue paths coincide, you can read off the corresponding root of the polynomial.[br][br]For instance, you can verify that the initial polynomial [math]f\left(x\right)=x^3+x^2-4x-4[/math] has real roots -1, -2, and 2. [br][br]Usability note: to get fine-grained control of the slider (on a desktop computer), click on the slider. Then use the left and right arrow keys to move it.