Has there ever been a time when you have had to multiply a binomial by itself, let’s say two or three or even four times?
Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher)
Sure, lots of times, right?
Well, if you’re anything like me, the more I multiply (i.e., FOIL) the likelihood that I make a silly mistake increases… which makes me grumpy.
Wouldn’t it be nice if there was a formula for the expansion of a binomial that is raised to any positive integer that could give us a pattern for finding the binomial expansion quickly?
Introducing the Binomial Theorem or Binomial Expansion!
With the power of Factorials, which is nothing more than multiplying a sequence of descending natural numbers, and our knowledge of Series, we can find Binomial Coefficients by following a really simply pattern or formula.
We will begin by looking at how to find Binomial Coefficients, which uses the the idea of mathematical Combinations, which is a way to choose k objects from a group of n objects.
Imagine we have a bowl of let’s say 20 apples, and we want to pick 5 of them. That’s the same thing as saying we want to pick 5 objects from a group of 20 objects!
Okay, but what does this do for us?
Binomial Expansion
Well, by the power of Combinations we can find the coefficients for any binomial raised to any positive integer!
This will lead us directly to the Binomial Theorem which is a way to rewrite a Binomial in Summation (Series) Notation, where all we have to do is plug in values and simplify!
Together we will look at six examples of the Binomial Expansion in detail to ensure mastery, and see that it definitely simplifies our work when multiplying out a binomial expression that is raised to some large power, as Purple Math so nicely states.
Then we will see how the Binomial Theorem generates Pascal’s Triangle.
Pascal’s Triangle is an array of numbers, that helps us to quickly find the Binomial Coefficients that are generated through the process of Combinations. It’s an awesome visual tool and will definitely simplify your work.
Binomial Theorem – Video
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