Part I

How would you simplify ? One way would be to multiply the binomial by itself three times. That would be very time consuming and prone to error. Surely, there must be an easier way! You will be happy to know that there is. The famous mathematician Blaise Pascal employed such a method in 1653, called his arithmetical triangle. The first account of his method was printed in 1665. To employ this method, you must construct a triangle as follows:

The entry on a particular cell of the triangle is formed by adding the two numbers directly above the cell. Then the numbers along a row are used to determine the coefficients of each term in the simplification of (that is, any power of a binomial). The top row is the coefficient of , the row below that yields the coefficients for , and so forth. Thus,

since the first row contains {1}.

since the second row contains {1, 1}.

since the third row contains {1, 2, 1}.

since the forth row contains {1, 3, 3, 1}.

Note that the exponents of x and y in each term sum to the power that you are expanding. So always start with x to that power as the first term, and then reduce the exponent of x by one and increase the exponent of y by the one for each subsequent term of the expansion.

One thing you should know for sure: 

1. Write the following power expansion, using the remaining rows of the triangle.

a.    b.    c.    d.

2. Now add two more rows to the triangle. Then use these to write the power expansion of 

Pascal's triangle will also work for other binomial power expansions. For example, if we replace y by -1,

3. See if you can use Pascal's triangle to determine the following binomial powers:

a.     b.    c.    d.    e.

Part II

Many Internet sites present interesting facts about Pascal's triangle. For example, if you were to take all the cells containing odd numbers and color them one color, and all the cells containing even numbers and color them another color, interesting patterns emerge. Search the Internet for a site that studies Pascal's triangle. Try to find a site that contains a large triangle grid, and print it out. If the cells of the triangle are not completely filled out, complete the calculations for the cells. The color all the even-numbers cells one color.

1. One of the patterns you should observe is symmetry. Write a short explanation of what this means to you.

2. Another pattern that is present is something called a fractal. Research this term, and explain what it means in relation to your triangle. Visit the companion website and obtain a program called Sierpins, that will create a fractal on your calculator.

Part III

Pascal's triangle was studied by others beside Pascal. Omar Khayyam was said to have studied it. Many centuries earlier, it was described by a Chinese mathematician, Yang Hui. Similar triangular arrangements were known to the Arabs about the same time the Chinese were using it. Find a reference on the history of mathematics, and report on a civilization that was known to use Pascal's triangle. Document your references. Identify any individuals, and describe what is known about the civilization's use of the triangle.