Introduction to Binomial Expansion | Finding a Specific Term with Binomial Expansion |

Expanding a Binomial | More Practice |

## Introduction to Binomial Expansion

The **Binomial Theorem** or **Binomial Expansion **(or **Binomial Series**) is a formula used to expand binomials like $ {{\left( {x+y} \right)}^{n}}$ into a sum with terms $ a{{x}^{b}}{{y}^{c}}$, where $ b$ and $ c$ are non-negative integers, and $ b+c=n$. A **perfect square trinomial** is a simple example: $ {{\left( {x+y} \right)}^{2}}={{x}^{2}}+2xy+{{y}^{2}}$. (The coefficients in this case are **1**, **2**, and **1**, respectively.)

It turns out that the coefficient $ a$ in the terms $ a{{x}^{b}}{{y}^{c}}$ of this expansion is equal to $ \left( {\begin{array}{*{20}{c}} n \\ c \end{array}} \right)$ (also written as $ \displaystyle {}_{n}{{C}_{c}}$), where $ \displaystyle \left( {\begin{array}{*{20}{c}} n \\ c \end{array}} \right)=\frac{{n!}}{{c!\left( {n-c} \right)!}}$; this is called the **binomial coefficient**. Remember that $ n$ factorial, or $ n!=n\left( {n-1} \right)\left( {n-2} \right)\,\,\,….$ (until you get to **1**). You can also get $ \displaystyle {}_{n}{{C}_{c}}$ on your graphing calculator. Type in what you want for $ n$, then **MATH, PROB**, and hit **3** or scroll to **nCr**, and then type $ c$ and then **ENTER** (or go directly to **nCr** and type in both numbers). $ \displaystyle {}_{n}{{C}_{c}}$ is actually the number of ways to choose $ c$ items out of $ n$ terms, where order doesn’t matter – also called the **Combination **function that we also saw here in the **Introduction to Statistics and Probability** section.

Thus, here is the formula for the **Binomial Theorem **(also called the** Binomial Formula **or** Binomial Identity**):

See how the exponents of the $ x$’s are going down (from $ n$ to **0**), while the exponents of the $ y$’s are going up (from 0 to $ n$)? And remember that **anything raised to the 0** is just **1**. Note that for a binomial raise to the “$ n$”, there are “$ n+1$” terms.

### Pascal Triangle

The coefficients can also be found using a **Pascal Triangle**, which starts with **1**, and is a triangle with all **1**’s on the outside. On the inside, the next number down is the sum of the two numbers above:

As an example of how to use the Pascal Triangle, the **3**^{nd} row shows the coefficients for $ {{\left( {x+y} \right)}^{2}}={{x}^{2}}+2x+{{y}^{2}}$, with coefficients **1 2 1**. Note that when using the Pascal Triangle, the exponent of the binomial is off by **1**; for example, we used the **3**^{nd} row to get the coefficients for $ {{\left( {x+y} \right)}^{2}}$.

Here’s another illustration of just how Pascal’s Triangle is used for expanding binomials:

Here’s a hint: when finding the coefficients of a binomial expansion using Pascal’s triangle, find the line with the **second term** the same as the power you want. For example, for a binomial with power **5**, use the line 1 **5** 10 10 5 1 for coefficients.

## Expanding a Binomial

The best way to show how Binomial Expansion works is to use an example. Expand $ {{\left( {x+3} \right)}^{6}}$ using the formula above. Here, the “$ x$” in the generic binomial expansion equation is “$ x$” and the “$y $” is “**3**”:

Notice how the power (exponent) of the first variable starts at the highest ($ n$) and goes down to **0** (which means that variable “disappears”, since $ {{\left( {\text{anything}} \right)}^{0}}=1$). Also notice that the power of the second variable starts at **0** (which means you don’t see it), and goes up to $ n$.

Also notice that for the coefficients of the $ \left( {\begin{array}{*{20}{c}} {} \\ {} \end{array}} \right)$ part, the **6** (since this is $ n$) always stays on top, and the bottom starts with **0** and goes up to **6**. The exponents for the first term of the binomial start with **6** ($ n$) and goes down to **0**, and the exponent on the second term is always the bottom part of the $ \left( {\begin{array}{*{20}{c}} {} \\ {} \end{array}} \right)$. If you add the two exponents, you always get **6**, which is $ n$.

Again, for the binomial coefficient $ \displaystyle \left( {\begin{array}{*{20}{c}} n \\ c \end{array}} \right)$, you can just use the $ \displaystyle {}_{n}{{C}_{c}}$ on your graphing calculator. (Type in what you want for $ n$, then **MATH, PROB**, and hit **3** or scroll to **nCr**, and then type $ c$ and then **ENTER**). You can also do these “by hand” by using $ \displaystyle \left( {\begin{array}{*{20}{c}} n \\ c \end{array}} \right)=\frac{{n!}}{{c!\left( {n-c} \right)!}}$. Notice that $ \displaystyle \left( {\begin{array}{*{20}{c}} n \\ 0 \end{array}} \right)\,\,\,\text{and}\,\,\,\left( {\begin{array}{*{20}{c}} n \\ n \end{array}} \right)$ is just **1** ($ 0!=1$) and $ \displaystyle \left( {\begin{array}{*{20}{c}} n \\ 1 \end{array}} \right)\,\,\,\text{and}\,\,\,\left( {\begin{array}{*{20}{c}} n \\ {n-1} \end{array}} \right)$ is $ n$.

To use the **Pascal Triangle** above to do this, look at the **7**^{th} row (since the first row is just “**1**”) to get the coefficients: 1 6 15 20 15 6 1. Note that since we want the **6**^{th} power, use the line that has **6** as the second term!

Here are a few more that are a little more complicated, including one that’s a **Complex Number**:

## Finding Specific Terms with Binomial Expansion

You may be asked to find **specific terms** using the Binomial Expansion; for example, they may ask to find the **5**^{th} term of a binomial raised to an exponent, or the term containing a certain variable raised to a power.

To do these, just remember that the $ x$^{th} term has $ (x-1)$ in the bottom of the $ \left( {\begin{array}{*{20}{c}} {} \\ {} \end{array}} \right)$ part of the **binomial coefficient**, since the first term has the $ \left( {\begin{array}{*{20}{c}} n \\ 0 \end{array}} \right)$ part. The $ x$^{th} term’s coefficient of a binomial expanded to the $ n$^{th} term is $ \left( {\begin{array}{*{20}{c}} n \\ {x-1} \end{array}} \right)$. The exponent of the first part of the expanded terms is the difference of the two numbers in the $ \left( {\begin{array}{*{20}{c}} {} \\ {} \end{array}} \right)$, and the exponent of the second part of the expanded terms equals the bottom number in the $ \left( {\begin{array}{*{20}{c}} {} \\ {} \end{array}} \right)$, since the two exponents always add up to equal $ n$.

For example, to expand a binomial raised to the **5**^{th} power, the **4**^{th} term has a $ \left( {\begin{array}{*{20}{c}} 5 \\ {4-1} \end{array}} \right)=\left( {\begin{array}{*{20}{c}} 5 \\ 3 \end{array}} \right)$ coefficient, the power of the first expanded term **5 – 3 = 2**, and the power of the second is **3**.

Here are some examples. Also remember that sometimes you will see $ {}_{n}C{}_{r}$ instead of $ \left( {\begin{array}{*{20}{c}} n \\ r \end{array}} \right)$:

**Learn these rules, and practice, practice, practice!**

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On to **Introduction to Limits** – you are ready!