Geometric Representation of (A+B)2=A2+2·A·B+B2
We want to construct a geometric figure that represents the algebraic identity (A+B)2=A2+2·A·B+B2. To start with, we want to represent the value A. We do this by drawing a line segment and stating that the length of the segment is 'A'.
We also want to represent the value B, so we draw a line segment and state that the length of the second segment is 'B'.
Click on the dots in the figure at left, hold down the mouse button, then drag them around to see how the figure changes.
Now we need to represent the value A+B. We do this by drawing line segment A and line segment B with a common end point and parallel to each other. The length of the combined line segment is A+B.
Click on the middle dot and move it. This shows that the length of the two segments is A+B.
Now we use the segment A to build a square. Each of the sides has the same length. The length of each side is 'A'. So the area of the square is A·A or A2. Click on the dot on the middle of the line and move it. Notice the sides of the square change to the same size as the segment A. The area of the square changes too.
Now we extend the lines. The new rectangle has an area of A·B.
Here, the figure is complete. Each side of the large square has a length of A+B. So the size of the large square is (A+B)2. There are four smaller rectangles within the square. One is of area A2, two have an area of A·B, and one has an area B2. Add these together you get A2+2·A·B+B2. But, since the area of the four rectangles is the same as the area of the square, we can conclude that (A+B)2=A2+2·A·B+B2.
Now consider the geometric representation of the algebraic identity (A+B)3=A3+3A2B+3AB2+B3. When we built the representation of (A+B)2, we ended up with a square. So, when we build the representation of (A+B)3, we will get a cube.
The diagram to the left is transparent, so you can see all the lines of the cube. First try to visualize the entire cube. Move some of the points by clicking on them and holding down the mouse button. This will help you visualize the entire cube. See if you can pick out one A3, three A2B, three AB2, and one B3.
Here is a printable document that will let you build a 3-D representation of (A+B)3=A3+3A2B+3AB2+B3: apb3.pdf.