The Binomial Theorem is a fundamental formula in mathematics that allows us to expand expressions of the form (a plus b) to the power of n, where n is any non-negative integer. The theorem states that this expansion equals the sum from k equals 0 to n of binomial coefficients times a to the power of n minus k times b to the power of k. The binomial coefficient is calculated as n factorial divided by k factorial times n minus k factorial. For example, when n equals 2, we get a squared plus 2ab plus b squared.
Binomial coefficients are the key to understanding the binomial theorem. These coefficients can be arranged in Pascal's Triangle, where each number is the sum of the two numbers above it. The first row contains just 1, the second row has 1, 1, the third row shows 1, 2, 1, and so on. Each row corresponds to the coefficients for expanding (a plus b) to successive powers. The recursive formula shows that any binomial coefficient equals the sum of the two coefficients above it in the triangle.
Now let's see how to expand (a plus b) cubed step by step. First, we apply the binomial theorem formula with n equals 3. This gives us the sum from k equals 0 to 3 of binomial coefficients times a to the power 3 minus k times b to the power k. Next, we write out each term explicitly: binomial 3 choose 0 times a cubed times b to the zero, plus binomial 3 choose 1 times a squared times b, and so on. Then we calculate the binomial coefficients: 1, 3, 3, and 1. Finally, we get a cubed plus 3 a squared b plus 3 a b squared plus b cubed.
Let's look at more examples of binomial expansions. When n equals 0, we get simply 1. For n equals 1, we have a plus b. We already saw n equals 2 gives a squared plus 2ab plus b squared. For n equals 4, the expansion has 5 terms with coefficients 1, 4, 6, 4, 1. And for n equals 5, we get 6 terms with coefficients 1, 5, 10, 10, 5, 1. Notice that all these coefficients come directly from Pascal's Triangle, demonstrating the beautiful connection between the binomial theorem and this famous mathematical pattern.
The Binomial Theorem has numerous applications across mathematics and science. It's fundamental in probability theory for calculating combinations, essential in combinatorics for counting problems, useful in calculus for series expansions, and important in physics and engineering for approximations. Let's summarize the key points: the general formula shows how any binomial raised to a power can be expanded using binomial coefficients from Pascal's Triangle. The expansion always has n plus 1 terms, where powers of a decrease while powers of b increase. Interestingly, the sum of all coefficients equals 2 to the power n. The Binomial Theorem truly is a powerful tool that connects algebra, combinatorics, and many areas of mathematics, making it one of the most elegant and useful formulas in mathematics.