| 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 3nd 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 3nd 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 7th 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 6th 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 5th 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 5th power, the 4th 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!