Completing the Cube

Date: 11/28/96 at 14:07:59
From: Stephan Smolka
Subject: Completing the cube

Dear Sirs,

I am familiar with the method of completing the square.  Could you
please show me the method of completing the cube or for that matter, 
a general method for all higher degree polynomials?

Thanks.

Date: 11/29/96 at 11:02:07
From: Doctor Jerry
Subject: Re: Completing the cube

There may be several possible answers to this question. However, I 
think it is fair to say that there is no widely known or used method 
for completing the cube.

From the history of mathematics it is clear that completing the square 
began as a geometric operation.  Evidence of this can be found in 
Greek, Mesopotamian, and Vedic civilizations.  One starts with a 
rectangle with unequal sides.  Let x be the shorter side and x+b the 
longer side.  Draw a rectangle with x vertical and x+b horizontal.  
In the left side of the rectangle draw a square of side x.  Divide the 
b by x strip on the right side of the rectangle into two equal parts 
by drawing a vertical line.  Move the rightmost thin strip to the 
bottom of the rectangle, turning it 90 degrees.  Align it under the 
x by x square.  You now have a square with a missing corner square.  
The missing square has dimensions b/2 by b/2.  We then must add 
b^2/4 = (b/2)^2 to complete the square.  All of this is echoed in the 
algebraic procedure:
 
  x^2 + bx = x^2 + bx + b^2/4 -b^2/4 = (x+b/2)^2 - b^2/4.

One can find diagrams in Cardan's work on the cubic that are analogous 
in three dimensions to the above geometry for completing the square. 
Just as completing the square is (algebraically) based on the 
identity:

           (x+a)^2 = x^2 +2ax + a^2

Cardan's constructions are based on:

           (x+a)^3 = x^3 +3x^2 a + 3xa^2+a^3

Thinking (for convenience) of a as quite small relative to x, if one 
adds three "slabs," each of volume 3ax^2, to three contiguous faces of 
a cube of side x, and then fits in three "butter sticks" of volume 
3xa^2, one has, except for a missing small cube (of volume a^3), a 
perfect cube.

Calculus students see an echo of this constuction when they compare 
the value of the increment (x+h)^3 - x^3 with the differential 3x^2h.

Completing the cube is more complex than completing the square. 
Moreover, to complete the square algebraically, we add and subtract a 
number free of x; we cannot, in general, complete the cube by adding 
and subtracting a number free of x.

If you want to learn more about Cardan, take a look at:

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cardan.html   

I hope this helps!   

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/