(I'm not sure why the spacing looks terrible today--I'm sorry!) Today's concept requires a basic understanding of binomials, FOILing, factoring, and Pascal's Triangle. For a great refresher on these topics, please see Khan Academy. This concept appeared 12 times last year, with a median placement at question # 54.
Number Dojo Level: 296
First, let me introduce (or re-introduce) you to Pascal's Triangle, in which each term in a given row is the sum of the two terms above it. This triangle becomes very useful in many areas of mathematics:
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
and so on.
For purposes of binomial expansions, the top row (with the lonely 1) is considered Row 0. That makes the (1 1) Row 1, the (1 2 1) Row 2, etc. Also, each row starts with 1, which is the 1st term, etc.
Finding the Coefficient of the Nth Term of a Binomial Expansion
The first concept we need to understand is that the order of the binomial expansion is the exponent that the binomial is being raised to. In other words, it will be the highest exponent in the resulting expansion. This order (or highest exponent) corresponds to the relevant row in Pascal's Triangle. So if a binomial is squared, the row we're concerned about is the (1 2 1) row, which is row 2.
To determine the coefficient of the nth term, we have to take the original coefficients to the same power as the variables they're attached to, and multiply that by the associated term in the Pascal's Triangle row. This is best seen with examples, so here we go:
- To find the sum of the coefficients of a binomial expansion, simply add the coefficients in the expansion and take that sum to the power (order) of the expansion.