1.9 Pascal's Triangle and the Binomial Theorem

Binomial Theorem

https://www.youtube.com/watch?v=iOQjV0FB9nY

1) HOW TO START A BINOMIAL EXPANSION: If your binomial is something like (x + 3) raised to a power like (x + 3)^5, you have two parts of your binomial: x and 3. You're going to take each of those and raise them to different powers in each term of the expansion. You can start your expansion just by writing these powers out.. For the FIRST PART of your binomial, x, it will start with a power of 5 (your power number) in the first term of the expansion (x^5), and then in each term after the power will go down by 1 (x^4, x^3, etc. all the way down to a zero power, x^0). Then you take the SECOND PART of the binomial, 3, and multiply each term by a power of 3. The 3 factor will start with a power of 0 in the first term, so 3^0, and then will increase by 1 power in each term after. 

2) HOW TO FIND THE COEFFICIENTS (with the FACTORIAL or COMBINATION method): Then there's one more number to find, a number that gets multiplied in front of each term, or a binomial coefficient. There are two ways to find the coefficients (for the Pascal's Triangle way, see below). To find the coefficients using the factorial, combination ("n choose k") formula of n!/(k!(n-k)!, each term has a coefficient number you find using an n value equal to the power number, 5, and a k value that runs from 0 for the first term up through 5 for the last term. This number gets multiplied by the other factors in each term, and then simplify for your final expansion. 

3) HOW TO FIND THE COEFFICIENTS (with PASCAL'S TRIANGLE): You can instead use Pascal's Triangle to find the binomial coefficients. Whichever row of Pascal's Triangle has your power number in it, as the second number, is the row that gives you all the coefficient numbers you'll need for your expansion. Each coefficient is multiplied with its term, and then you can simplify the expansion. 4) WHAT IF THE BINOMIAL HAS SUBTRACTION? It's very similar. It's easier to think of the subtracted term as "adding a negative number", and then all of the negative number (in parentheses) will be raised to the power in each term of the expansion. Note: if the binomial have something like 2x - y instead of x - y, make sure that all of (2x) is raised to each power.