Pascal's triangle
Pascal's triangle is an array of numbers that represents a number pattern. It has a number of different uses throughout mathematics and statistics, but in the context of polynomials, specifically binomials, it is used for expanding binomials.
Properties of Pascal's triangle
Below is a portion of Pascal's triangle; note that the pattern extends infinitely.
- Pascal's triangle is symmetrical; if you cut it in half vertically, the numbers on the left and right side in equivalent positions are equal.
- The outermost diagonals of Pascal's triangle are all "1."
- The values inside the triangle (that are not 1) are determined by the sum of the two values directly above and adjacent. Refer to the figure below for clarification.
In the figure above, 3 examples of how the values in Pascal's triangle are related is shown. From top to bottom, in yellow, the two values are 1 and 1, which sums to 2, the value below. Similarly, 3 + 1 = 4 in orange, and 4 + 6 = 10 in blue. - The sum of the numbers in each row of Pascal's triangle is equal to 2^{n} where n represents the row number in Pascal's triangle starting at n=0 for the first row at the top. Refer to the figure below for clarification.
There are other properties of Pascal's triangle aside from those listed above, but understanding those listed above can be useful when using Pascal's triangle to expand binomials.
Binomial expansion
Pascal's triangle can be used to identify the coefficients when expanding a binomial. Specifically, the binomial coefficient, typically written as , tells us the b^{th} entry of the n^{th} row of Pascal's triangle; n in Pascal's triangle indicates the row of the triangle starting at 0 from the top row; b indicates a coefficient in the row starting at 0 from the left.
Example
Expand: (x + y)^{3}
(x + y)^{3} = | |
= |
Refer to the following figure along with the explanation below.
In this example, n = 3, indicates the 4^{th} row of Pascal's triangle (since the first row is n = 0). The numbers in the row, 1 3 3 1, are the coefficients, and b indicates which coefficient in the row we are referring to. Refer back to the example above. The coefficient on the first term, x^{3}, is that in b = 0 of row n = 3, or 1. The same follows for each corresponding term such that the coefficient of the 2^{nd}, 3^{rd}, and 4^{th} terms are 3, 3, and 1 respectively, exactly as in row n = 3 of Pascal's triangle.
Next, we can determine the values of the expressions multiplied by each coefficient. Each term has some component of x and some component of y raised to an exponent. The exponent on the x and y components sum to n. Starting from the left, x has an exponent equal to n, or 3, and y has an exponent of 0. Moving from left to right, 1 is subtracted from the exponent on the x component while 1 is added to the exponent on the y component, which results in the final term having an exponent of 0 on the x component, and an exponent of 3 on the y component. This can be seen in the example above, where the exponents on each term are explicitly written.
Using Pascal's triangle and these patterns, we can expand binomials raised to nth powers that would otherwise be very tedious to expand through repeated multiplication. Refer to the binomial theorem page for the formulaic approach to expanding binomials, which is even more efficient once you are comfortable with all the mathematical symbols in the formula.