The binomial theorem can be a tedious theorem to implement that's why today on Math Hacks
I am showing you the techniques that I use to simplify the process and stay organized in
Today's tutorial you'll learn not only what the binomial theorem is
You'll also learn about some quick and easy checks
You can use to avoid errors and a shortcut that will greatly reduce the amount of computations you need to make.
When you crack open your textbook or do a quick google search for the binomial theorem
You're going to stumble across a formula that it's a bit confusing the first time you see it
So we have a lot going on here in this formula we have Sigma notation to represent
Summations we have the choose formula from combinatorics
we have all sorts of exponents and variables and
factorials going on and
You're thinking this looks like a lot of work just to multiply some binomials together
But bear with me it will pay off in the end. Plus I'm going to show you some shortcuts to help you get through
What could be a really long problem. So let's go ahead and do an example
Let's expand X minus 3y to the fifth
So what this means is that I want to multiply X minus 3y with itself five times
And if you've ever multiplied out
Binomials, you know that for the first two or three binomials you multiply together. It's not too bad
But as we add on this fourth and fifth and so on it can get pretty nasty
And that's something we really don't want to do. So that's why we have this formula
So to apply this formula
The first thing I want to do is figure out what my a and b are so a is the first term
So in my binomial here a will be X. So I'm just gonna write a
Underneath it
And then my B is going to be
Negative 3y and it's important that you take that negative sign
Along with you when you do this expansion
The third thing I want to look for is the power
so the N in this case, my n is 5
so what we've found, is that a
equals x
B equals negative 3y n
Equals 5 so for the first term in our binomial expansion, I'm going to start with K of 0
that's what this Sigma notation here means and I'm going to plug in these values for the rest of my variables and
Use 0 for K. So the first term is going to look like
So what I have here is 5 choose 0
Multiplied with X to the 5 minus 0 power and negative 3 y to the 0 power
Notice how I have the parentheses here and have the exponent on the outside
that is a really important step because you want to make sure that this exponent is being applied to both the
coefficient and the variable now what we would do next is go through and
increment K to 1
So now I'm going to do the exact same thing and plug these in except for put 1 for K
Since we're going to have 6 of these terms you can already see that
This is gonna take a couple rows if I start writing it
horizontally, so what I want to do here just to stay organized is I'm going to write each term in a column as if they
Were mini problems and then I'm just going to add together all of the answers to those miniature problems
So below this I'm going to write out the second term
So again, these values are going to stay the same and I'm going to use a K of 1 this time
Now you may notice a little pattern here
We have 5 minus 0 in this first one on the exponent which is 5 and we have 5 minus 1 here
Which is 4 so what's going to happen here is
The first term as we go through is going to count down from the degree
Which is 5 in this example
So it's going to count down from 5
- 4 - 3 - 2 - 1 - 0 and our second term is going to count up from 0 to 5
You may also notice that the K value is going to be
incrementing each time by one so instead of going through this process of
Entering everything into this formula each time. It's really easy to make mistakes that way. So what I would recommend doing here is
writing out
the parenthesis
Placing the 5 the X and the negative 3y for the rest of the terms
so you're always going to have one more term than the
Degree of what you're expanding.
So since this is 5 I'm going to have 6 terms so I'm going to just quickly write out four more of these below it
Okay, so now that I've copied down the parts that are gonna remain the same and all of these little miniature problems I can go
ahead and
Increment my numbers. So let's start with the K value
So the K starts with a zero, and then it goes to one. This one will be two
three four and five
And you should end up where the last two numbers are the same
Now let's go ahead and do the powers on X. So I started with five
For this one's going to be three
two
One and zero and you should always end with zero in this column
and then for the last one for negative three why we're counting up from zero, so we have 0 1
2 3 4 and 5
And what I like about doing it this way is it prevents me from making some really silly
Transcription errors when I try to just write it all out
one at a time another quick check that you can do just to make sure that you haven't made any errors is
The powers on your two terms should always add to the power that you're working with
So 5 plus 0 is 5
4 plus 1 is 5
3 plus 2 is 5 2 plus 3 is 5 1 plus 4 is 5 and 0 plus 5 is 5
Now that we have outlined all of the little miniature problems. We need to solve the next thing to do is to simplify them
So we're going to solve each of these
problems
So I promised you shortcuts and I have another really important shortcut here
So the next thing we need to do is take care of each of these little problems
now the first part of the problem is to
Solve 5 choose 0 and that comes from our combinations formula
Which we have outlined up here and what we would do is we would plug in 5 for n and 0 for K
So we'd have 5 factorial over
5 minus 0 factorial times 0 factorial then we would go through and apply the factorials cross-cancel
And get an answer and if that is the way that your teacher is requiring you to solve these problems
Then I would definitely do it that way
If you need a little more explanation of the combinations formula, I have another tutorial on that.
But today I'm talking all about shortcuts. So the shortcut that you can use here is instead of using the
Combinations formula which our theorem tells us were supposed to use we can use Pascal's triangle
So the first thing I need to show you is what Pascal's triangle is now
It's really easy to make what you do is you start with a 1 and then below it you write
Two other ones and on the outside of each row
You're going to write a 1 so we'll put another set of ones here and then in between I'm going to just add together
Add together the two numbers above it
So I get 2 let's do another row set up 1 on the ends and then I would add
Like this so 1 plus 2 is 3 and 2 plus 1 is 3 it's again another row I'd do ones
Pascal's formula is really awesome because it tells us exactly
The numbers we would get if we went and solved each of these choose formulas
so how it works is you're going to make your Pascal triangle and you're gonna keep doing these rows until you have
the same number of rows as
Terms, so see here. I had 1 2 3 4 5 6 terms
So I've made 1 2 3 4 5 6 rows and it is this six row that we're going to be working with another
Way to look at this is that if you are
Expanding to a power of five then it's going to be one more than that number
Now all we have to do is match these up with our formula. So 5 choose 0 is 1
Just cross this out and put 1 isn't that nice and in five choose one is going to be 5
5 choose 2 is 10
5 choose 3 is 10
5 choose 4 is 5
and 5 choose 5 is 1
So we've already taken care of skipping out on all of this math that we'd have to do and being able to directly evaluate those
The next thing I want to do is go ahead and simplify anything to the zero power
Because the zero power always equals one. So this just becomes 1 and down here. This just becomes 1
Alright, so things have already gotten a lot easier now. I'm just going to go through each of these problems and simplify them down
So for the first one
I have 1 times X to the fifth times 1 so that's just going to be X to the fifth and
The second problem I have
5x to the fourth
times negative 3y
And I can multiply the 5 and the negative 3 together. So this one turns out to be negative
15 X to the 4th Y
The third problem, I have 10 X cubed negative 3y squared
So write that
so here is going to be 9y squared when I apply that power of 2, so I have 10 times 9 which
Is 90 X cubed Y squared
For the next one again. I have 10
x squared and
negative 3y cubed
When I multiply out the negative 3 3 times with itself, I get negative 27 Y cubed
So negative 27 times 10 gives me negative
270 x squared Y cubed
Five
x to the first power and
Negative three Y to the fourth
so both of those with the fourth, I could write it like that and
when I multiply out negative three four times with itself, I get positive 81, so this becomes
5x
times 81
Y to the fourth and then oh this this is kind of big one
We have five times 81, of course
You could pull out your calculator or you could go ahead and do a little mental math trick here
we have five times eighty plus five times 1 so 5 times 80
Oh, that sounds kind of hard but when you think about it
It's really just 5 times 8 times 10, right because 8 times 10 is 80. So 5 times 8 is 40
Times 10 is 400 plus that 5 times 1 so we get 405
And for our last term here
We have 1 the X went away times negative 3 y all to the fifth
so all I need to do here is
Multiply out negative 3 5 times and for that one. You might want to pull out a calculator
When you insert in the calculator you get negative
243 Y to the 5th. All right, so we have found all of the terms and
Our last job here is just to sum them all together
I
Went ahead and wrote in plus signs in between all of the terms
but of course where I have a negative coefficient
I could have just written a minus sign here. And that's the best way to write the answer
So I'm just going to quickly do that
And it's always a good idea to double check everything before you finish
so make sure that all of your powers on each term add to
the degree that you started with, so five in this case and when I do that
I see that I already made a transcription error. So on this one, I have x squared Y squared that only equals four
So I know right away that I made a mistake and yep when I look back here. It looks like I wrote down a
Squared here instead of a three
So easy to do on these type of problems, so you always want to check
Let's fix that
All right. So now checking it. I have 5
4 & 1 3 & 2 2 & 3
1 & 4 & 5
Looks like it all checks out
For more Math Hacks and shortcuts make sure to click the subscribe button below and if you learned something new today
Let me know by giving this video a big thumbs up
Until next time I'm Brett and thank you so much for joining me
For more infomation >> Os vários sabores de Linux - android? Como assim? - Duration: 9:21. 
For more infomation >> Seat Ibiza New 1.0 EcoTSI 95pk Style | Navi | Stoelverwarming | - Duration: 1:08. 
For more infomation >> Volkswagen Transporter | 150 | Navigatie | Parkeersensoren | Lange Wielbasis - Duration: 1:05. 
For more infomation >> Seat Ibiza 1.0 EcoTSI 95pk 5drs Style Connect Navi - Duration: 1:06. 
For more infomation >> Sam Bee On TBS
For more infomation >> ¡A Barack Obama le gusta el ritmo de J Balvin! | Suelta La Sopa | Entretenimiento - Duration: 0:31. 
For more infomation >> Win a Free Car at Circle K | SA Live | KSAT 12 - Duration: 3:32. 
For more infomation >> Fiat Panda 1.2 Emotion - Duration: 1:01. 
No comments:
Post a Comment