Relations Between Roots and Coefficients

Transcript

0:00Introduction

Hello everyone! First, one small challenge. Alpha and beta are the roots of x square minus five x plus three equals zero. Find alpha square plus beta square. Most students find the roots, square them, add them, and two minutes are gone, right? Today you will do it in seconds, without finding the roots at all. And look, in JEE, questions of this type come up often. So, let's begin.

0:30Sum and product of the roots

So, take any quadratic, a x square plus b x plus c equals zero, with roots alpha and beta, these two Greek letters. Now tell me: if you know the roots, how can you write the quadratic itself? Yes, exactly! As factors. The quadratic equals a, times x minus alpha, times x minus beta. Put x equal to alpha, and the right side is zero, right? Same for beta. Clear? Now multiply out the right side: a x square, minus a times alpha plus beta, times x, plus a alpha beta. Both sides are the same quadratic. So the coefficients must match. Compare the x terms first. On the left, b; on the right, minus a times alpha plus beta. So alpha plus beta is b over a, right? No, no, careful! The minus sign stays. Sum of the roots: alpha plus beta equals minus b over a. Dropping this minus sign is a classic JEE mistake. Don't do it! Now the constant terms: c equals a alpha beta. So the product, alpha beta, is c over a, no minus sign this time. And both relations hold even when a, b and c are complex. Always!

1:45Symmetric expressions: build them from s and p

Now, short names: s for the sum, and p for the product. Expressions like alpha square plus beta square stay the same when you swap alpha and beta; we call them symmetric. And we build each one from s and p. Okay? First, alpha square plus beta square. The sum is s, so the sum of the squares is just s square, right? No, no! Look, s square is alpha plus beta, whole square: alpha square plus beta square, plus two p. The double arrow means implies: alpha square plus beta square equals s square minus two p. Makes sense? Cubes the same way. s cube has an extra three p s inside, so alpha cube plus beta cube equals s cube minus three p s. Write it down! Reciprocals? Common denominator. One over alpha plus one over beta is alpha plus beta, over alpha beta: s over p. Really simple! Now the difference. Alpha minus beta, whole square, is alpha square plus beta square minus two p: s square minus four p. That also equals D over a square, where D is the discriminant. Take the square root; that is the root sign. Mod of alpha minus beta, with vertical bars for modulus, equals root D over mod a. The brackets say: only when D is greater than or equal to zero. Got it?

3:14Power sums: Newton's recurrence

Now, let's go to bigger powers. Alpha to the n plus beta to the n, for a large n? Expanding is a nightmare, right? Call it S n; the small n below is a subscript. Here is the trick. Alpha is a root, so a alpha square plus b alpha plus c is zero. The arrow says: multiply by alpha to the n minus two. We get a alpha to the n, plus b alpha to the n minus one, plus c alpha to the n minus two, equals zero. Same for beta. Add the two, and that's it: a S n, plus b S n minus one, plus c S n minus two, equals zero, for n greater than or equal to two. This is Newton's recurrence. Clear? But to start the chain, we need S zero and S one. S zero is alpha to the zero plus beta to the zero. So S zero is zero, right? Oh no! Alpha to the zero is one, and beta to the zero is one, so S zero is two. Write zero here, and everything after goes wrong. And S one is just the sum: minus b over a. With me?

4:21Topper's trick: complex roots? Go polar

Now a topper's trick: complex roots? Go polar. For x square minus two x plus two equals zero, the roots are one plus minus iota; the plus minus sign gives both. In polar form, the e to the power form from complex numbers: root two, e to the plus minus iota pi by four; pi is this Greek letter. So alpha to the n plus beta to the n is two, times two to the n by two, times cos of n pi by four; the dot means multiply. Any power, in one line, no recurrence needed! Okay? And x square plus x plus one has roots omega and omega square; x square minus x plus one has minus omega and minus omega square. Omega is our cube root of unity, so use omega cube equals one. Super quick!

5:11Example 1: sum of squares, cubes, reciprocals of the roots

Now, remember our challenge? Let's finish it. Alpha and beta are the roots of x square minus five x plus three equals zero. Find alpha square plus beta square, alpha cube plus beta cube, one over alpha plus one over beta, and mod of alpha minus beta. Pause the video and try. Done? Okay, let's check. Here a is one, b is minus five, and c is three. So s, minus b over a, is five, and p, c over a, is three. Now our challenge. Alpha square plus beta square is s square minus two p: twenty five minus six, nineteen. See, challenge solved in seconds, and we never touched the roots! Cubes: s cube minus three p s. Five cube is one hundred and twenty five, three times three times five is forty five. So, eighty. Easy, isn't it? Reciprocals: s over p, five by three. And mod of alpha minus beta is root of, s square minus four p: root of twenty five minus twelve, root thirteen. Four answers, not a single root found!

6:23Example 2: power sums by Newton's recurrence (JEE Main 2020)

Now a harder one, JEE Main twenty twenty. Alpha and beta are the roots of x square minus x minus one equals zero, and p k is alpha to the k plus beta to the k; here p k is a power sum, not the product. Which statement is not true? Pause and try. Hint: use the recurrence. Done? Okay, let's check. Here a is one, and b and c are minus one. So Newton gives: p n, minus p n minus one, minus p n minus two, equals zero. Each term is the sum of the two before it. Start the chain: p zero is two, not zero; remember the trap. And p one is minus b over a, that is one. Then keep adding the last two: three, four, seven, eleven. Got it? Now the options; a tick means true. Option a: p five minus p four is eleven minus seven, four, and that is p three. Option b: p five is eleven. Option c: p one plus, the dots mean and so on, up to p five: one plus three plus four plus seven plus eleven, twenty six. Now option d: p five equals p two times p three. Alpha to the five is alpha square times alpha cube, so this must be true too, right? No, no, careful! That is the trap. p two times p three is three times four, twelve; the slashed equal sign means not equal to eleven. Multiplying two sums brings extra cross terms. So option d is not true: our answer.

8:02Quick revision: sum, product, symmetric forms, power sums

So, quick revision. Sum is minus b over a, product is c over a; watch that minus sign. Symmetric expressions: write them in s and p, no need to find the roots. Power sums: a S n plus b S n minus one plus c S n minus two equals zero, starting from S zero equals two. Now go and try the practice problems on this page. All the best!