The Binomial Theorem

Transcript

0:00Introduction

Hello everyone! Okay, a quick puzzle before we start. Take two x plus three, the whole thing to the power four. What is the coefficient of x cubed? Many students answer in two seconds: four, because four C one is four, right? Hmm, is it really? By the end of this video you will know, and you will never multiply four brackets by hand again. Expansions like this come up often in JEE, so let's begin.

0:31Where the coefficients come from

So, where do the numbers in an expansion come from? Take x plus a, the whole cubed. It is just three brackets of x plus a, multiplied together. Okay? Now, how is each term made? From every bracket you pick either x or a, and multiply your three choices. The tree on the right shows this: the box at the top is our three brackets, and each level below is one bracket, with an x or an a in every circle. Two choices at each bracket, so eight paths in all, and under each path, the term it makes. With me? Now look at the bottom of the tree. Below the orange line, the tree counts the paths for each term, and x squared times a comes from three paths. But why exactly three? Tell me. Yes, exactly! For x squared times a, exactly one bracket gives a, and the other two give x. So we only choose which bracket gives the a: three C one, three ways. The C with a small three at the top left and a small one at the bottom right means: choose one bracket out of three. Clear? And the same counting gives every term. x plus a, the whole cubed, equals three C zero x cubed, plus three C one x squared a, plus three C two x a squared, plus three C three a cubed. The same line is in orange under the tree. The coefficient of each term is simply the number of ways to choose which brackets give the a. Right?

2:03The binomial theorem

Now, the same reasoning works for any power, with one condition: n is a natural number, that is, a positive integer. x and a can be any numbers, real or complex. Got it? Then x plus a, to the power n, equals n C zero x to the power n, plus n C one x to the power n minus one times a, plus n C two x to the power n minus two times a squared, and so on, the three dots, up to n C n a to the power n. The power of x falls by one each time, and the power of a rises by one. Okay? In short: the big Greek letter sigma means add up, for r from zero to n. The term with a to the power r is n C r, x to the power n minus r, a to the power r, with r counted from zero. Clear? Now, quick question. In x plus a, to the power n, how many terms are there? n terms? No, no, careful! r runs from zero to n, so there are n plus one terms. That plus one catches many students. And in every term, the powers of x and a add up to n. With me? The numbers n C zero up to n C n are the binomial coefficients. Terms at equal distance from the two ends have equal binomial coefficients, because n C r equals n C n minus r. Right?

3:30Expansions with a minus sign

Now, what about x minus a, to the power n? It is just a replaced by minus a. So what happens to each term? Minus a, to the power r, gives minus one to the power r, times a to the power r. So x minus a, to the power n, is the same sum with minus one to the power r in front. Even r gives plus, odd r gives minus: the signs alternate. Okay? So tell me, what is the sign of the last term, the one with a to the power n? Always minus? Oh no, careful! There r equals n, so the sign is minus one to the power n: plus when n is even, minus when n is odd. Got it? Two special cases you will use all the time. When the first term is one, its powers are all one, so one plus x, to the power n, is simply the sum of n C r, x to the power r. Clear? And one minus x, to the power n, is the same sum with minus one to the power r in front. With me?

4:34Example 1: expand (2x + 3)^4

Alright, now our puzzle from the start. Expand two x plus three, to the power four. Pause the video and try it. Done? Okay, let's check. The binomial coefficients, four C zero up to four C four: one, four, six, four, one. Symmetric, as expected. Right? Now each term. Two x, the whole to the power four, plus four times two x the whole cubed, times three, plus six times two x the whole squared, times nine, plus four times two x, times twenty seven, plus eighty one. Nine is three squared, twenty seven is three cubed, and eighty one is three to the power four. The powers of two x fall, the powers of three rise. With me? Now our puzzle. The x cubed term has r equal to one, so its binomial coefficient is four C one, which is four. So the coefficient of x cubed is four, right? No, no, careful! That is the trap. Four is only the binomial coefficient. The coefficient of x cubed also takes the numbers inside the brackets: four, times two cubed, times three. The dots mean times. Ninety six! Not four. Puzzle solved. Clear? So the full answer: sixteen x to the power four, plus ninety six x cubed, plus two hundred and sixteen x squared, plus two hundred and sixteen x, plus eighty one. Perfect!

6:07Example 2: expand (x^2/2 - 3/x)^4

Now a harder one. Expand x squared over two, minus three over x, the whole thing to the power four. Fractions and a minus sign, together. Pause the video and try it. Done? Okay, let's check. Same binomial coefficients: one, four, six, four, one. And with the minus sign, the signs alternate: plus, minus, plus, minus, plus. Right? Now the terms. First, x squared over two, to the power four: x to the power eight over sixteen. Second: four, times x squared over two, the whole cubed, which is x to the power six over eight, times minus three over x in brackets. Third: six, times x to the power four over four, times minus three over x, the whole squared, which is nine over x squared. Okay? Fourth: four, times x squared over two, times minus three over x, the whole cubed, which is minus twenty seven over x cubed. And last, minus three over x, the whole to the power four: eighty one over x to the power four, a plus sign, because the power is even. Got it? Now you simplify the second term: four, times x to the power six over eight, times minus three over x. Minus three over two, times x to the power five. Correct! In the same way, the third term is twenty seven over two, times x squared, and the fourth is minus fifty four over x. So the answer: x to the power eight over sixteen, minus three over two times x to the power five, plus twenty seven over two times x squared, minus fifty four over x, plus eighty one over x to the power four. Very good!

8:06Quick revision: n+1 terms, alternating signs, coefficients

So, quick revision. x plus a, to the power n, is the sum of n C r, x to the power n minus r, a to the power r: n plus one terms, and the powers add up to n. With a minus sign inside, the signs alternate. And the coefficient of a power of x is n C r times the numbers inside the brackets. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!