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Binomial Theorem

The capital sigma you see in the definition is called the summation symbol. This is typically read as "the sum from k equals 0 to n." With this theorem, we can start writing out the first few powers of n of the binomial.

(x + y)^0 = 1

(x + y)^1 = C(1,0) * x^(1-0) * y^0 + C(1,1) * x^(1-1) * y^1 = x + y

(x + y)^2 = x^2 + 2xy + y^2

Now, I'll write the next few by hand.

This is just to n = 5. But with computers, we can go up to numbers beyond that like n = 50 or n = 100! However, there's something interesting that lies within these coefficients. There's a pattern in there, but can you see it? If not, maybe it will help if I write only the coefficients in the following way.

Now can you see it? Don't worry if you're not able to see it. A key thing to note is that everywhere along the edge of the triangle is a 1 (i.e. 1 starts and ends each row of the triangle). The next term is created by adding the two terms above it. I'll show a diagram below of what I mean. 

After realizing this quick pattern, so long as you remember the first two rows of the triangle, you can find the binomial coefficients without needing to solve out factorials. This triangle has a special name and it's called "Pascal's Triangle." The triangle is actually infinite if you continue the pattern forever, but in most cases that you will see, you should be able to write out the triangle yourself in a small corner of the page. I know I did this on an exam to the power of 7 and the teacher laughed when she saw a part of the page was covered in Pascal's Triangle. I got full points on the problem, so it did help. I can number out the what number n each layer stands for, which I find is best remembered by looking at the second number in each row.

As you can probably see, each power, n, lines up with the second number in each row (with the exception of the first row).

Now, I'll do a few examples that will help to apply the binomial theorem to more than just (x + y). For now, we'll stick to low n's, but the same process can be applied to higher powers.

Ex. 1: Expand (x+3)^3

First, try this yourself. Then look at the solution after you've finished.

SOLUTION:

Ex. 2: Expand (2x + 3y)^2

SOLUTION:

Another thing the Binomial Theorem is useful for is for finding specific terms in the expansion of a binomial. There is a formula for this, which will be given in the next example.

Ex. 3: Find the 17th term in the expansion of (7a + 3b)^21 

The formula for the kth term of the expansion of (x + y)^n is C(n, k-1) * x^(n-(k-1)) * y^(k-1).

Try solving this problem out first by hand (use a calculator to multiply large numbers together to save some time) and then check the solution once you're finished.

SOLUTION:

Yeah.... In hindsight I kind of regret choosing this as an example. On the plus side, if you can do this by hand, whatever your teacher throws at you won't be nearly this horrible.

This covers the Binomial Theorem, I hope this has helped some of you in understanding what the Binomial Theorem is and how to use it. 

Sources:

Binomial theorem. (2021, January 06). Retrieved February 01, 2021, from https://en.wikipedia.org/wiki/Binomial_theorem

Combination. (2021, January 30). Retrieved February 01, 2021, from https://en.wikipedia.org/wiki/Combination