11.6 Binomial theorem

A polynomial with two terms is called a binomial. We have already learned to multiply binomials and to raise binomials to powers, but raising a binomial to a high power can be tedious and time-consuming. In this section, we will discuss a shortcut that will allow us to find ${(x+y)}^n$ without multiplying the binomial by itself $n$ times.

Identifying binomial coefficients

In Counting Principles , we studied combinations . In the shortcut to finding ${(x+y)}^n,$ we will need to use combinations to find the coefficients that will appear in the expansion of the binomial. In this case, we use the notation $\left(\begin{array}{c}n\\ r\end{array}\right)$ instead of $C(n,r),$ but it can be calculated in the same way. So

The combination $\left(\begin{array}{c}n\\ r\end{array}\right)$ is called a binomial coefficient . An example of a binomial coefficient is $\left(\begin{array}{c}5\\ 2\end{array}\right)=C(5,2)=10.$

Using the binomial theorem

When we expand ${(x+y)}^n$ by multiplying, the result is called a binomial expansion    , and it includes binomial coefficients. If we wanted to expand ${(x+y)}^{52},$ we might multiply $(x+y)$ by itself fifty-two times. This could take hours! If we examine some simple binomial expansions, we can find patterns that will lead us to a shortcut for finding more complicated binomial expansions.

First, let’s examine the exponents. With each successive term, the exponent for $x$ decreases and the exponent for $y$ increases. The sum of the two exponents is $n$ for each term.

Next, let’s examine the coefficients. Notice that the coefficients increase and then decrease in a symmetrical pattern. The coefficients follow a pattern:

These patterns lead us to the Binomial Theorem , which can be used to expand any binomial.

Another way to see the coefficients is to examine the expansion of a binomial in general form, $x+y,$ to successive powers 1, 2, 3, and 4.

Can you guess the next expansion for the binomial ${(x+y)}^5?$

See [link] , which illustrates the following:

• There are $n+1$ terms in the expansion of ${(x+y)}^n.$
• The degree (or sum of the exponents) for each term is $n.$
• The powers on $x$ begin with $n$ and decrease to 0.
• The powers on $y$ begin with 0 and increase to $n.$
• The coefficients are symmetric.

To determine the expansion on ${(x+y)}^5,$ we see $n=5,$ thus, there will be 5+1 = 6 terms. Each term has a combined degree of 5. In descending order for powers of $x,$ the pattern is as follows: