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One of the most important concepts in mathematics is the Binomial Theorem, especially as we approach more advanced subjects like calculus and precalculus. We may already be aware of what binomials are. If this is the case, then we're already halfway toward understanding the significance of the Binomial Theorem. But how exactly do we use the Binomial Theorem? Why is it so important? Let's find out:
Before we get into the Binomial Theorem, let's review the foundational concept: Binomials themselves.
A binomial is a polynomial with two different terms. Recall that a polynomial is an expression with many terms. You may also recall that a polynomial can have only positive exponents. Polynomials cannot have:
Note that while dividing by a variable is not allowed, you can divide variables by constants. So while $\frac{1}{x}$ is not allowed, $\frac{x}{2}$ is allowed.
A binomial is simply a type of polynomial that has two and only two terms. There are also monomials (one term), trinomials (three terms), and so on.
Here is an example of a binomial:
As we can see, there are two terms here:
AndThe Binomial Theorem answers a very simple question: What happens when we multiply a binomial by itself many times over? In other words, what happens when we take $5{x}^{2}-3$ and add an exponent to this entire expression? We would get something like this:
Now here's the real question: How do we expand an expression like this? One obvious solution is to distribute the expression. Let's take an easy example:
How did we get this result? We may recall the FOIL method of polynomial multiplication, which states the order of our operations:
Note that in our original example, we had an exponent of 3 not 2. To see how this works, let's multiply our result by $\left(a+b\right)$ once again:
This leaves us with:
We can even multiply this result by $\left(a+b\right)$ once again, giving us:
Although it might not be obvious at first, a pattern is starting to emerge here. For example, look at the result of ${\left(a+b\right)}^{3}$ :
Focus only on the exponents of a. We see a pattern of 3, 2, 1, 0. We can use this pattern to make a prediction with the Binomial Theorem. In other words, this theorem maps the expansion of ${\left(a+b\right)}^{n}$ , allowing us to swap in n for any exponent and find our answer relatively simply. We can write this pattern as:
Here $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ represents "n choose k", which is a binomial coefficient calculated as:
This is the number of ways to choose k elements from a set of n elements
Each term in the expansion corresponds to a term in the sequence of binomial coefficients, multiplied by the corresponding powers of x and y. The powers of x decrease from n to 0 across the terms, while the powers of y increase from 0 to n.
If you're feeling a little lost, don't worry. We're about to break down these expressions in greater detail. Just remember that x is the first term in the binomial expression, while y is the second term. We should also remember that "!" indicates a factorial number, which means we need to multiply all whole numbers from the chosen number down to one. For example, $4!=4\cdot 3\cdot 2\cdot 1=24$ . We also know that n is the exponent that we're using to multiply $\left(a+b\right)$ . Understanding the pattern only gets us so far, and we still need a method that gives us the coefficients. We can do this with the formula for the binomial theorem:
In this formula:
The Binomial Theorem has a few interesting properties:
Like many complex mathematical concepts, we can graph the pattern of the Binomial Theorem, and this visual representation may be easier to understand for visual learners. If we try to graph the results of a binomial distribution with a simple graph, we will soon find out that the resulting pattern is always symmetrical. It climbs to a peak before declining once again while remaining perfectly symmetrical.
If we created a more advanced visual representation of this concept, we would be left with something called "Pascal's Triangle." This is also called "Yang-Hui's Triangle" in many areas of the globe. This number triangle shows us the Binomial Theorem. Specifically, it shows us the distribution of coefficients in increasing powers of binomial expansion.
This pattern isn't just an interesting oddity -- it can also be very useful to memorize. If we understand how binomial distribution works, we can rely on our memory of the patterns -- as laid out in Pascal's Triangle. This is often much more straightforward than actually solving the Binomial Theorem.
Let's use our newfound skills to solve a problem that involves the Binomial Theorem:
What is the coefficient of ${a}^{4}$ in the expansion of ${\left(1+a\right)}^{8}$ ?
In other words, what is the number that lies next to ${a}^{4}$ when we expand ${\left(1+a\right)}^{8}$ ?
We can use a variation of the Binomial Theorem to find our answer:
The general term of the expansion of ${\left(x+y\right)}^{n}$ is $\frac{n!}{\left(n-r\right)!r!}{x}^{\left(n-r\right)}{y}^{r}$
Where:
Here:
Although this formula might seem daunting at first, we already know the values we need to plug in:
Now all we need to do is plug those terms into our equation:
Now we can do our factorial operations:
Now we know that the coefficient of ${a}^{4}$ in the expansion is 70.
Writing Number Patterns in Function Notation
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