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Difference of Two Cubes: x^3 – y^3 = (x-y)(x^2+xy+y^2)

how.PNG Difference of Two Cubes: x^3 – y^3 = (x-y)(x^2+xy+y^2)
p1.PNG Difference of Two Cubes: x^3 – y^3 = (x-y)(x^2+xy+y^2)
p2.PNG Difference of Two Cubes: x^3 – y^3 = (x-y)(x^2+xy+y^2)
p3.PNG Difference of Two Cubes: x^3 – y^3 = (x-y)(x^2+xy+y^2)
X3-Y3_Blocks.png Difference of Two Cubes: x^3 – y^3 = (x-y)(x^2+xy+y^2)
X3-Y3_Case.png Difference of Two Cubes: x^3 – y^3 = (x-y)(x^2+xy+y^2)
X3-Y3_Blocks.stl Difference of Two Cubes: x^3 – y^3 = (x-y)(x^2+xy+y^2)
X3-Y3_Case.stl Difference of Two Cubes: x^3 – y^3 = (x-y)(x^2+xy+y^2)
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  • X3-Y3_Blocks.stl
  • X3-Y3_Case.stl

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Publication date 2020-04-08 at 06:57
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Published to Thingiverse on: 2020-04-08 at 05:31
Design number 131206
LGBU

LGBU

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3D model description

####x^3 –y^3 = (x-y)(x^2+xy+y^2)

Physical Model for the Difference of Two Cubes

This set of 3D blocks demonstrate, physically, the well-known algebraic fact that x^3 – y^3 = (x-y)(x^2+xy+y^2). While it is possible to show that the equation is true in a physical sense, it is a bit of a stretch toward physical modeling. The algebraic manipulation might be more aesthetically appealing. That is why we still need algebra.

The bid ideas covered are: (1) If the area of a polygon is (x^2+xy+y^2) square units, then, when extended (or extruded) for (x-y) units in the 3rd dimension, the resulting solid has a volume of (x-y)(x^2+xy+y^2); (2) the three solids x^2(x-y), xy(x-y), and y^2(x-y) can be arranged into a x^3 minus y^3, as shown in the pictures.

A box is also included for help with the arrangement of the blocks for the (x^2+xy+y^2) base. Of course, it does not matter how these three pieces are laid out on the floor as long as one can identify the three terms, x^2, xy, and y^2. By the way, what is the volume of the interior of the box?

Have fun!

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