Binomial Expansion

This section covers:

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):

Binomial Theorem

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.

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:

Pascal Triangle 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 2nd 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”:

Binomial Expansion

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 7th 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 “4a” and the “y” is “–3b”:

Binomial Expansion 2

We also could have used the 5th 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 5th 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 xth  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 xth  term’s coefficient of a binomial expanded to the nth 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 5th power, the 4th 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   \(\left( {\begin{array}{*{20}{c}} n \\ r \end{array}} \right)\):

Specific Term of a Binomial Expansion
Learn these rules, and practice, practice, practice!

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On to Introduction to Limits – you are ready!

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