Lill's method

[2] A later paper by Lill dealt with the problem of complex roots.

[3] Lill's method involves drawing a path of straight line segments making right angles, with lengths equal to the coefficients of the polynomial.

The sequence of directions (not turns) is always rightward, upward, leftward, downward, then repeating itself.

The final point reached, at the end of the segment corresponding to the equation's constant term, is the terminus.

For every real zero of the polynomial, there will be one unique initial angle and path that will land on the terminus.

A quadratic with two real roots, for example, will have exactly two angles that satisfy the above conditions.

, ... are successively generated as distances between the vertices of the polynomial and root paths.

When a path is interpreted using the other convention, it corresponds to the mirrored polynomial (every odd coefficient's sign is changed), and the roots are negated.

When the right-angle path is traversed in the other direction but with the same direction convention, it corresponds to the reversed mirrored polynomial, and the roots are the negative reciprocals of the original roots.

[4] Lill's method can be used with Thales's theorem to find the real roots of a quadratic polynomial.

A circle is drawn with the straight line segment joining the start and end points forming a diameter.

Intersects of this circle with the middle segment of Lill's method, extended if needed, thus define the two angled paths in Lill's method, colored blue and red.

The negative of the gradients of their first segments, m, yield the real roots 1/3 and −2.In 1936, Margherita Piazzola Beloch showed how Lill's method could be adapted to solve cubic equations using paper folding.

[7] In this example with 3x3 + 2x2 − 7x + 2, the polynomial's line segments are first drawn on a sheet of paper (black).

The axis of reflection (dash-dot line) defines the angled path corresponding to the root (blue, purple, and red).

The negative of the gradients of their first segments, m, yield the real roots 1/3, 1, and −2.

Finding roots −2, −1 (repeated root), and −1/3 of the quartic 3 x 4 +13 x 3 +19 x 2 +11 x +2 using Lill's method. Black segments are labeled with their lengths (coefficients in the equation), while each colored line with initial slope m and the same endpoint corresponds to a real root.
Finding roots −1/2, −1/ √ 2 , and 1/ √ 2 of the cubic 4 x 3 + 2 x 2 − 2 x − 1 , showing how negative coefficients and extended segments are handled. Each number shown on a colored line is the negative of its slope and hence a real root of the polynomial.
Finding roots of 3 x 2 + 5 x − 2
Find roots of 3 x 3 + 2 x 2 − 7 x + 2