The Binomial Theorem

Hi everyone! My name is Maridel and today I am going to summarize what we learned in class today which is... 
The Binomial Theorem.

We learned how to expand and simplify using binomial expansion.
 
Firstly, let us expand and simplify the following:








Do you notice any patterns?
As you can see, when you look at the exponent for each binomial and +1, that is the number of terms when the binomial is expanded and simplified. 
So for example, if







 

The exponent in this case is zero. Zero plus one is equal to one and therefor you end up with one term.

Also, when expanded and simplified you will notice that this value of exponents for x decreases as the value of the exponents for y increases.
For example, if

 

The exponents of x decrease each term. As you can see the exponent of x goes from 6, to 5, to 4, to 3, and so on.. While the exponents of y increases each term. The exponent of y starts off at 0. Then it goes to 1, to 2, to 3 and so on..


The coefficients also follow a pattern! If you compare pascal's triangle to 
On the left, is Pascal's triangle. As you can see, the expanded and simplified binomials follow the same pattern. In row 1 we start off with 1. In the next row according to pascal's triangle there are two 1's and the picture on the left justifies pascal's triangle because the coefficient of both x and y are 1. You may also notice the hockey stick pattern! The numbers that are highlighted created a hockey stick shape. When the numbers on the long end of the hockey stick are added together, the sum is equal to the number on the short end!
 
Next, let me introduce the formula:

 

Let's try an example now, shall we?
Expand and Simplify using the binomial expansion:





First things first, start off by listing the values: 
a= 3x       b= -y      n=5 ... n is equal to 5. Just remember that in each term, the exponents must add up to the value of n. And in this case, the exponents in each term must add up to 5! So whether the exponents are x to the power of 1 and y to the power of 4, or x to the power of 2 and y to the power of 3.. as long as they add up to five you are doing perfectly fine!!
*NOTE: Don't forget the negative signs!

 Next, use the formula and plug in the values. By doing this you are expanding.


  
And now, lets Simplify! Remember that anything to the power of 0 is equal to 1!





And we are left with:



I recommend that you always, always, ALWAYS double check your answers.

... And that's it! Hopefully you now all understand the concept of binomial expansion. Have a great day! and don't forget...