This section covers:

**Introduction to Binomial Expansion****Expanding a Binomial****Finding a Specific Term with Binomial Expansion****More Practice**

# Introduction to Binomial Expansion

You’ll probably have to learn how to expand polynomials to various degrees (powers) using what we call the **Binomial Theorem** or **Binomial Expansion **(or **Binomial Series**).

We use this when we want to expand (multiply out) a the power of a binomial like \({{\left( {x+y} \right)}^{n}}\) into a sum with terms \(a{{x}^{b}}{{y}^{c}}\), where *b* and *c* are nonnegative integers (and it turns out that *b* + *c* = *n*). Let’s see a really simple example: \({{\left( {x+y} \right)}^{2}}={{x}^{2}}+2xy+{{y}^{2}}\). (The coefficients in this case are 1, 2, and 1, respectively.)

It just turns out that the coefficient *a* in this expansion is equal to \(\left( {\begin{array}{*{20}{c}} n \\ c \end{array}} \right)\) (also written as \(\displaystyle {}_{n}{{C}_{c}}\)), where \(\left( {\begin{array}{*{20}{c}} n \\ c \end{array}} \right)\,\,=\,\,\frac{{n!}}{{c!\left( {n-c} \right)!}}\) (this is called the **binomial coefficient**). Remember that \(n!=n\left( {n-1} \right)\left( {n-2} \right)\,\,\,….\) (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). \(\displaystyle {}_{n}{{C}_{c}}\) is actually the number of ways to choose (order doesn’t matter) *c* items out of *n* terms – also called the **Combination **function.

So here is the **Binomial Theorem** (it’s really not as bad as it looks):

See how the exponents of the ** x**’s are going down (from

*n*to 0), while the exponents of the

*’s are going up (from 0 to*

**y***n*)? And remember that

**anything raised to the 0**is just

**1**.

Actually, 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. Then on the inside, add the two numbers above to get the next number down:

As an example of how to use the Pascal Triangle, start with the second row for \({{\left( {x+y} \right)}^{1}}=1x+1y\), so the coefficients are both 1. When using the Pascal Triangle, the exponent of the binomial is off by 1; for example, we used the 2^{nd} row to get the coefficients for \({{\left( {x+y} \right)}^{1}}\).

# Expanding a Binomial

The best way to show how Binomial Expansion works is to use an example. Let’s 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 with 6 (*n*) and goes down to 0, and the exponent on the second term is always the bottom part of the \(\left( {\begin{array}{*{20}{c}} {} \\ {} \end{array}} \right)\). And if you add the two exponents, you always get 6 (since this 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 \(\left( {\begin{array}{*{20}{c}} n \\ c \end{array}} \right)\,\,=\,\,\frac{{n!}}{{c!\left( {n-c} \right)!}}\). 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 always 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 just *n*.

To use the **Pascal Triangle** above to do this, let’s look at the 7^{th} row (since the first row is just “1”) to get the coefficients: 1 6 15 20 15 6 1.

Let’s try another expansion, that’s a little more complicated. Here, the “*x*” in the generic binomial expansion equation is “4*a*” and the “*y*” is “–3*b*”:

We also could have used the 5^{th} row of the Pascal Triangle to get the coefficients.

Notice how every other term is negative, since the second term of the binomial is negative.

# 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 say 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. (So the *x*^{th} term’s coefficient of a binomial expanded to the *n*th term is \(\left( {\begin{array}{*{20}{c}} n \\ {x-1} \end{array}} \right)\).)

Then remember that 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, if we are expanding a binomial raised to the 5^{th} power, the 4^{th} term will have 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. And 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!