even wanting everyone today we're going to talk about Pascal's triangle we're going to use it for something called binomial expansion what if we had to expand X plus 5 to the fourth well in the past that would be X + 5 X + 5 X plus 5 X plus 5 and you have to distribute out the entire thing well with Pascal's triangle it's something called the binomial theorem we can use this triangle to create the expression that means the same thing now real quick how to build the triangle the ones are on the outside so once on the outside ones on the outside the number here is you get when you add these two so you get three three four six four and then we'll just keep going the next line would be 1 5 10 10 1 so how does this work well you have two terms in here an A and a B notice that the powers of a go down and then it's the powers of the B go up and notice that the coefficients matched the correct line of Pascal's triangle it's easier when you see an example so let's try this problem x + 5 to the fourth well let's go back to Pascal's triangle this is the zero power the first power second power the third power the fourth power so we're going to use this line 1 4 6 4 1 there's are going to be the coefficients that we use for right now 1 1 one of the two terms that I have one of them is X so I'm going to put an X X X X X the other term is five powers powers when the X the first term start the biggest power which is four and they go down powers are the five the second term started the zero and go up you can always double check yourself because the sum of the two powers has to equal the original power so 4 plus 0 is 4 3 plus 1 is 4 2 plus 2 is 4 now that we've expanded it let's simplify it this one's really easy this is just an extra 4 plus 4 times 5 is 20 X to the 3rd plus 5 squared is 25 times 6 would be 150 x squared plus 5 to the third is 125 125 times 4 would be 500 X plus 5 to the fourth well I might be calculated out by out to the fourth 625 much faster than if I had to boil everything out again because fishings come from Pascal's triangle the X's and the files come from the two terms in the binomial the powers you have down on the first term and go up on the secretary let's try one is a little bit more complicated I want the fifth power Pascal's triangle so it's this line as I did here a second ago 1 5 10 10 5 1 supposed to coefficients one the eighth term is x 3 X 3 X 3 X 3 X 3 X 3 X 3 X and the B term is negative for y now why does the negative matter because if I raise a negative to an even power it's going to stay and become positive if I raise it to an odd power it's gonna stay negative this seems like a lot but it's still a lot faster what would happen ecology this should be disaster rate these are these with good power there's a lot of time so what are the powers going to be well we start with the a value the biggest power is five four three two one zero the B value start with zero and we go up again you can always double check yourself because this is some of the two powers taxable or the power we started with so 4 plus 1 3 plus 2 2 plus 3 notice something every place that there was an odd power on this negative you can end up being - in between so the signs you're going to alternate so for the first one it's just 3 X to the fifth which is 243 X to the fifth I went ahead and it ahead of time so I didn't have to do in the board - because now I have a negative power 5 times 3 to the fourth times 4 is 1620 X to the fourth Y to the first this next term is going to be positive because it's negative squared so I'm going to be 10 times 3 to the 3rd times 4 to the second which is a positive 4320 X cubed Y squared the next term is going to be negative because it's negative to the 3rd power 10 times 3 squared times 4 to the 3rd 5760 x squared Y cubed 5 times 3 times negative 4 to the 4th were negative to where the fourth 20 positive 3840 I'm down to X to the 1st wild up dwarf amount of rooms right here slash term is going to be negative this is a non power this is one anything to the zero is one so then the last thing is just four to the fifth which is negative of 1024 X to the zero one by no means mansion using Pascal's triangle