2.3. Polynomial Expansion and Pascal’s Triangle http://www.ck12.org
(x+y)^0 = 1
(x+y)^1 =x+y
(x+y)^2 =x^2 + 2 y+y^2
(x+y)^3 =x^3 + 3 x^2 y+ 3 xy^2 +y^3Notice that the coefficients for thexandyterms on the right hand side line up exactly with the numbers from Pascal’s
triangle. This means that given(x+y)nfor any powernyou can write out the expansion using the coefficients from
the triangle. When you study how to count with combinations then you will be able to calculate the value of any
coefficient without writing out the whole triangle.
There are many patterns in the triangle. Here are just a few.
- Notice the way each number is created by summing the two numbers above on the left and right hand side.
- As you go further down the triangle the values in a row approach a bell curve. This is closely related to the
normal distribution in statistics. - For any row that has a second term that is prime, all the numbers besides 1 in that row are divisible by that
prime number. - In the game Plinko where an object is dropped through a triangular array of pegs, the probability (which
corresponds proportionally to the values in the triangle) of landing towards the center is greater than landing
towards the edge. This is because every number in the triangle indicates the number of ways a falling object
can get to that space through the preceding numbers.