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Binomial Expansion There was a famous French mathematician called Blaise Pascal, and he came up with this pattern called Pascalโs triangle, named after himself.
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So if we start with one, this is our first row, so weโll call this row ๐ equals zero.
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Now underneath one, Iโm gonna write two more ones.
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And weโve got these two numbers by adding the two numbers above them, so looking at the first one, weโve added one and zero together to get one.
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And then looking at the second one, the same thing.
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Weโve also added one and zero.
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Now this line is our ๐ equals one line.
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Looking at the next line, we can see exactly which numberโs gonna go underneath those two ones, because weโre gonna add them together and weโll get two.
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And then again on either side of the two, weโre adding zero on one, so weโre going to get one on either side of the two.
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This is the term ๐ equals two.
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And moving on to the next line, adding two and one together twice gives us three.
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You should by now have worked out whatโs gonna go on either side of the threes: one, good.
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So each time, weโre just adding the two numbers on top to give us the next number.
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So again moving down from our ๐ equals three to ๐ equals four.
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So first of all, weโre gonna have a one on either side, and then one add three is four, three add three is six, and three add one is four.
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This probably seems a bit strange right now, saying why are you just making a triangle and adding things above to go down rows?
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But you should be able to see that we could carry on going for infinity.
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Thatโs just up to row ๐; we could carry on going forever.
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Let me try and show you how weโre going to use it.
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One of the ways we use it is by expanding brackets.
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So if I have ๐ฅ plus ๐ฆ all to the power of zero, we know that anything to the power of zero is just gonna be one.
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How about ๐ฅ plus ๐ฆ to the power of one?
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Well thatโs gonna be ๐ฅ plus ๐ฆ.
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I wouldnโt usually but on this occasion Iโm going to write the coefficients of one.
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Okay letโs try ๐ฅ plus ๐ฆ all squared.
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We know thatโs the same as all of ๐ฅ plus ๐ฆ multiplied by all of ๐ฅ plus ๐ฆ, so letโs do it on the side.
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Multiplying out using FOIL, weโll do the first terms, so ๐ฅ multiplied by ๐ฅ gives us ๐ฅ squared.
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Outside is ๐ฅ multiplied by ๐ฆ, giving us ๐ฅ๐ฆ.
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Inside which is ๐ฆ multiplied by ๐ฅ, giving us ๐ฅ๐ฆ.
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And last which is ๐ฆ multiplied by ๐ฆ, giving us ๐ฆ squared.
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So we can collect those like terms giving us ๐ฅ squared plus two ๐ฅ๐ฆ plus ๐ฆ squared.
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And then if we write that down back where we originally were.
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Weโll do one more multiplication; weโre gonna have ๐ฅ plus ๐ฆ all cubed.
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So this we know is all of ๐ฅ plus ๐ฆ multiplied by all of ๐ฅ plus ๐ฆ multiplied by all of ๐ฅ plus ๐ฆ, but weโve already done ๐ฅ plus ๐ฆ all squared, so weโre just gonna take that and multiply it by ๐ฅ plus ๐ฆ again.
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So to do this, weโre going to take every term in the first set of parentheses and multiply them by ๐ฅ.
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And then weโre going to add on to that everything in the first set of parentheses multiplied by ๐ฆ.
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Okay now and if we collect the like terms, we can see weโve got two ๐ฅ squared ๐ฆ, and then weโve got another ๐ฅ squared ๐ฆ, so weโll have three ๐ฅ squared ๐ฆs.
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Weโve got one ๐ฅ๐ฆ squared and another two ๐ฅ๐ฆ squared.
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So adding that together, weโll get three ๐ฅ๐ฆ squared, and then finally a ๐ฆ cubed.
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Okay so then letโs write this back where the others are.
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So hopefully, you should have started to notice something.
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If we look at all the numbers in blue for our expansions and the numbers in the Pascalโs triangle, you should be able to see that theyโre exactly the same.
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And if you look at the powers for each of the expansions, the first one is to the power of zero.
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Remember I said thatโs where ๐ is equal to zero, that top line?
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And then the next one weโve got the power of expansion is one.
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Thatโs the line of ๐ equals one, and so on and so forth, so that ๐ actually talks about what weโre putting our parentheses to the power of.
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Now weโre gonna have a go.
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Iโm gonna show you what the binomial expansion theorem is, and we can have a look at how we would apply that and apply our knowledge now of Pascalโs triangle to help us expand really large brackets without having to go through all that hassle we just did then.
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So the binomial theorem is, ๐ plus ๐ all to the power of ๐ is equal to the sum of from ๐ equals zero to ๐ of ๐ choose ๐ multiplied by ๐ to the power of ๐ minus ๐ multiplied by ๐ to the power of ๐.
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This seems little bit confusing.
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So letโs actually show what that means.
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So it means ๐ choose zero multiplied by ๐ to the power of ๐ plus ๐ choose one multiplied by ๐ to the power of ๐ minus one multiplied by ๐ to the power of one plus every single term in between and then ๐ choose ๐ minus one all multiplied by ๐ to the power of one all multiplied by ๐ to the power on ๐ minus one plus ๐ choose ๐ multiplied by ๐ to the power of ๐.
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So in our calculators, we can put in ๐ choose any number, and itโs usually a button that looks like this, depending on which calculator you have.
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But if we donโt have a calculator, then we can either use Pascalโs triangle to give us the ๐ choose whatever weโre looking for or we can use factorials, where ๐ choose ๐ is equal to ๐ factorial all divided by all of ๐ minus ๐ all factorial multiplied by ๐ factorial.
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In this video, weโre going to focus on what would happen if we use our calculator and also Pascalโs triangle, but you can apply this, what weโve just shown there, where ๐ choose ๐ with factorials, to any one of the examples we do.
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Expand ๐ฅ plus two ๐ฆ all to the power of five using binomial expansion.
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Okay so letโs apply the expansion; weโll have five choose zero multiplied by ๐ฅ to the power of five plus five choose one multiplied by ๐ฅ to the power of four multiplied by two ๐ฆ to the power of one.
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So we can see here our powers in ๐ฅ are going to decrease by one each time, and our powers in ๐ฆ are going to increase by one each time.
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So then the next time, weโll have five choose two multiplied by ๐ฅ cubed multiplied by two ๐ฆ all squared plus five choose three multiplied by ๐ฅ squared multiplied by two ๐ฆ all cubed.
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Weโll have to go on another line for this.
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I always end up doing them with binomial expansion.
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Thatโs five choose four ๐ฅ to the power of one so ๐ฅ, all multiplied by two ๐ฆ to the power of four.
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And then our final term five choose five, now ๐ฅ because itโll be ๐ฅ to power of zero, and then multiplied by two ๐ฆ all to the power of five.
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So before we start calculating anything, letโs just have a look at what weโve done.
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We can see that each time weโve got ๐ is equal to five; then weโre choosing a number thatโs increasing each time until we get to five.
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We can see the ๐ฅ powers are decreasing by one each time and the ๐ฆ powers are increasing by one each time.
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Now having a go at calculating it, you can put in your calculator like I said using the button five choose zero, five choose one, et cetera.
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Iโm just gonna grab the fifth line from Pascalโs triangle, which if you remember the fourth line was one four six four one, but then adding the two numbers above, we will get one, five, ten, ten, five, and one.
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These will be the values of all of our ๐ choose ๐s individually.
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So first, we have one multiplied by ๐ฅ to the power of five, so thatโs just ๐ฅ to the power of five.
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Then five choose one which is five.
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Multiplying that by two ๐ฅ to the power of four and ๐ฆ.
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Then we can see the five choose two will be ten.
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Weโre gonna multiply that by two squared, two squared we know is four, and ๐ฅ cubed and ๐ฆ squared, and five choose three weโve got another ten.
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Multiplying that by two cubed, two cubed is eight, and then by ๐ฅ squared and ๐ฆ cubed.
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And five choose four we can see is five.
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Multiplying that by two to the power of four which is sixteen, and ๐ฅ and ๐ฆ to the power of four.
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Then for our last term, we can see five choose five is one, and weโre multiplying that by two to the power of five, which is thirty-two ๐ฆ to the power of five.
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Okay and then if we find out all the numbers here, weโve got ๐ฅ to the power of five plus ten ๐ฅ to the power of four ๐ฆ plus forty ๐ฅ cubed ๐ฆ squared plus eighty ๐ฅ squared ๐ฆ cubed plus eighty ๐ฅ๐ฆ to the power of four plus thirty-two ๐ฆ to the power of five.
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And we have expanded that bracket using binomial expansion, and that is a lot quicker than if weโd have had to do each of those multiplications out of the brackets individually.
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But what if in a question weโre not asked for the whole thing?
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What if weโre just asked for one term?
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What is the fourth term in the expansion two ๐ฅ minus three all to the power of eight?
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So first of all, think about the things we need to do.
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First of all, we need to do ๐ choose sum ๐.
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We know the ๐ will be eight, and the ๐ will be three.
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As if we remember that ๐ starts off on zero, so the fourth term would go zero one two three, so three would be the fourth term.
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This then is able to tell us the power of two ๐ฅ; weโre gonna put two ๐ฅ to the power of five because itโs two ๐ฅ to the power of eight minus three.
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And then we know itโs going to be negative three all to the power of three, as that is our ๐.
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So if you put eight choose three into a calculator, you get an answer of fifty-six.
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Youโre gonna multiply that by two to the power of five, which we know is thirty-two, and negative three to the power of three, which we know is negative twenty-seven.
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Of course, donโt forget ๐ฅ to the power of five.
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So then if we multiply these altogether in our calculators, we get a rather large number of negative forty-eight thousand three hundred and eighty-four ๐ฅ to the power of five.
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So there we have it.
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To find individual terms, we need to be sure of the ๐ and the ๐.
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To find the ๐, itโs just gonna be whatever power the bracket is to.
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So in this case, it is eight.
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And finally ๐ is gonna be one less than the term number we want.
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So in summary, we have used Pascalโs triangle and binomial theorem to help us carry out binomial expansion of some very large brackets.
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And binomial theorem, this formula, is specifically helpful for if we want to find just individual terms because if we ignore the sum thatโs exactly what weโll use.