A Given Power of x; the Term Independent of x

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle before we start. Take x minus one over x, the whole thing to the power ten. Expand it, and you get eleven terms. But in one of them, the x's cancel completely, and only a plain number is left. Which term is it, and what is that number? Will you write out all eleven terms? No way! Today we will do it in two lines. Questions like this come up often in JEE, so let's begin.

0:33Step 1: find the power of x in the general term

So, first recall the general term. T r plus one, capital T with a small r plus one written below, is the r plus one th term. It equals n C r, with the small n at the top left of capital C, times the first term to the power n minus r, times the second term to the power r. Here n is a positive integer, and r runs zero, one, up to n; the three dots mean and so on. We count from r equals zero, okay? Now take a general binomial: alpha x to the power p, plus beta x to the power q, the whole thing to the power n. Alpha and beta, the two Greek letters, are the numbers in front. The general term: n C r, times alpha x to the power p, raised to n minus r, times beta x to the power q, raised to r. With me? Now tell me. In this term, what is the total power of x? Yes, exactly! The power p, raised to n minus r, gives p times, n minus r. The power q, raised to r, gives q times r. When we multiply, the powers add: p times, n minus r, plus q r. Clear? So, for the coefficient of x to the power k, set this power equal to k, and solve for r. That's the whole method: one small equation in r. Right?

1:56The term independent of x: set the power to 0

Now, the term independent of x. It simply means the term with no x at all, the constant term. No x means x to the power zero, so k is zero. Got it? So let's solve our puzzle: x minus one over x, to the power ten. The general term: ten C r, times x to the power ten minus r, times, in round brackets, minus one over x, to the power r. See, I keep the minus sign inside the bracket. Okay? Now your turn. One over x is x to the power minus one. So what is the total power of x, and which r makes it zero? Correct! Ten minus r, then minus r again: ten minus two r. Set it equal to zero. The double arrow on screen means implies, so r equals five. With me? So r is five: that is T six, the sixth term, because we count from r equals zero. It is ten C five, times minus one to the power five. Ten C five is two hundred fifty two, and the sign is minus. So the term independent of x is minus two hundred fifty two. Puzzle solved, in two lines! Clear?

3:10Trap: when r is not a whole number

Now, a trap. Does x squared plus one over x, to the power ten, have a term in x to the power six? Don't rush to find a coefficient. First ask: does this term even exist? Let's check. The x squared, to the power ten minus r, gives two times, ten minus r. One over x, to the power r, gives minus r. Total: twenty minus three r. Okay? Set twenty minus three r equal to six. Three r is fourteen, so r is fourteen over three. Many students now put fourteen over three into ten C r and push on to some answer. Oh no, careful! Fourteen over three is not a whole number, and r counts the terms, so it must be an integer from zero to n. Right? See it directly. Put r equals zero, one, two, and so on. The powers go twenty, seventeen, fourteen, eleven, eight, five, two, and so on, dropping by three each time. Six is skipped! Got it? So there is no x to the power six term, and its coefficient is zero. Remember: if you get a fraction for r, do not force an answer out of it. The term simply does not exist. Clear?

4:25Example 1: coefficient of x^7 in (2x^2 - 1/x)^11

Alright, example one. Find the coefficient of x to the power seven, in two x squared minus one over x, the whole thing to the power eleven. Pause the video and try it. Done? Okay, let's check. General term: eleven C r, times two x squared, to the power eleven minus r, times minus one over x, to the power r. The minus sign and the two stay inside their brackets. Okay? Power of x: two times, eleven minus r, then minus r. That is twenty two minus three r. Set it equal to seven: three r is fifteen, so r is five. Five is a whole number between zero and eleven, so the term exists. With me? Now the coefficient: eleven C five, times two to the power eleven minus five, that is two to the power six, times minus one to the power five. Quick question. Positive or negative? Negative, yes! Minus one to an odd power is minus one. Eleven C five is four hundred sixty two, and two to the power six is sixty four. Four hundred sixty two times sixty four, the cross means times, is twenty nine thousand five hundred sixty eight. So the coefficient is minus twenty nine thousand five hundred sixty eight. Never drop that minus sign. Clear?

5:48Example 2: constant term in (sqrt(x/3) + 3/(2x^2))^10

Now a harder one. Find the term independent of x in: square root of x over three, plus three over two x squared, the whole thing to the power ten. The tick-shaped sign is the square root sign, okay? Pause the video and try it. Hint: root x is x to the power half. Done? Okay, let's check. General term: ten C r, times root of x over three, to the power ten minus r, times three over two x squared, to the power r. With me? Now, root of x over three, to the power ten minus r. What power of x does that part give? Yes, exactly! Root x is x to the power half, so we get ten minus r, over two. And three over two x squared, to the power r, gives x to the power minus two r. Set the total to zero: ten minus r, over two, minus two r, equals zero. So ten minus r equals four r, and r equals two. Okay? So r is two: the third term, T three. It is ten C two, times one third to the power four, times three over two, squared. Why one third to the power four? Root of one third, to the power eight, is one third to the power four. Right? Ten C two is forty five, and the small dots mean times. One third to the power four is one over eighty one, and three over two, squared, is nine over four. Forty five over eighty one is five over nine, and five over nine, times nine over four, is five over four. So the term independent of x is five over four. Same method, even with a square root!

7:33Quick revision: power of x, solve for r, the no-term trap

So, quick revision. One: write T r plus one, collect the power of x, and set it equal to k. Two: for the term independent of x, set the power equal to zero. Three: if r is not an integer in the closed interval zero to n, shown in square brackets, there is no such term, and the coefficient is zero. Four: keep the signs and the numbers inside the brackets. Now go and try the practice problems on this page. All the best!