Cubic Equations: Roots and Coefficients

Transcript

0:00Introduction

Hello everyone! So, one small puzzle first. Take this cubic: x cube minus six x square plus eleven x minus six equals zero. Its roots are one, two and three. Now look at the numbers in the equation: six, eleven, six. And look at the roots: one, two, three. Is there some hidden connection? Oh yes, there is! Today you will read the roots' secrets straight from the coefficients, without solving the cubic. And look, this idea is very useful in JEE problems. So, let's begin.

0:39A pattern hiding in the coefficients

Here is our cubic again: x cube, that small three up there means cube, minus six x square, the small two means square, plus eleven x, minus six, equals zero. And its roots: one, two and three. First, add the three roots. What do you get, and where have you seen that number? Six, yes! One plus two plus three is six. And the coefficient of x square? Minus six. The same six, only with the sign flipped. Interesting, right? Now multiply the roots in pairs, and add the three products. Then multiply all three together. What do you get? One times two, plus two times three, plus three times one: eleven, exactly the coefficient of x! The small dot here simply means times. And one times two times three is six: again our constant term, minus six, with the sign flipped. Coincidence? Not at all. It is a rule for every cubic. Clear?

1:43Vieta for a cubic

Now in general. Take any cubic: a x cube, plus b x square, plus c x, plus d, equals zero. Call its three roots alpha, beta and gamma: these three Greek letters, okay? The sum, alpha plus beta plus gamma, equals minus b over a, just like the quadratic. And the pair products, alpha beta plus beta gamma plus gamma alpha, equal plus c over a. Same pattern as our six and eleven. Got it? Now the product, alpha beta gamma. For a quadratic, the product was c over a, with a plus sign. So for the cubic, it is d over a, right? No, no, careful! This is a classic trap. The product is minus d over a. So the signs go minus, plus, minus: they alternate. With me? Now check our hook. Here a is one, b is minus six, c is eleven, and d is minus six. Minus b over a gives six, c over a gives eleven, and minus d over a gives six. Exactly our six, eleven, six! The tick mark means it checks out. Mystery solved.

2:57Build other sums without solving

Alright, now the real power. First, a short notation. This big Greek letter is sigma, and it means sum. Sigma alpha is alpha plus beta plus gamma, and sigma alpha beta is the sum of the three pair products. Now, alpha square plus beta square plus gamma square: sigma alpha square. Can you build it from these two? Square the sum, yes! When you square sigma alpha, every pair product comes twice, right? So subtract two sigma alpha beta. Sigma alpha square equals sigma alpha, the whole square, minus two sigma alpha beta. Next, one over alpha, plus one over beta, plus one over gamma. Take alpha beta gamma as the common denominator, and the top becomes beta gamma, plus gamma alpha, plus alpha beta. So sigma one over alpha equals sigma alpha beta, over alpha beta gamma. Really simple. And for cubes, one more for your notebook. Sigma alpha cube equals three alpha beta gamma, plus sigma alpha, times, sigma alpha square minus sigma alpha beta. Every piece on the right comes from Vieta. Clear?

4:11Topper's trick: roots in A.P. or G.P.

Now, a topper's trick. Sometimes the question tells you the roots are in A P, arithmetic progression. How would you name the three roots, to make life easy? Keep the middle one as a, and step d on either side: a minus d, a, a plus d. Why? Add them, and the d's cancel, so the sum is just three a, nothing else. And the sum, Vieta already gives us! Careful: this a and d are new letters, not the cubic's coefficients, okay? For G P, geometric progression, take a over r, a, and a r. Multiply them, and the r's cancel, so the product is just a to the power three. And the product also comes straight from Vieta. Perfect! So the plan: find a first, then d or r, in that order. One unknown at a time. That's it!

5:04Example 1: sum of squares and reciprocals of the roots

Now, Example one. Alpha, beta and gamma are the roots of x cube minus three x square plus two x minus one equals zero. Find sigma alpha square, and sigma one over alpha. Pause the video, and try it first. Done? Okay, let's check. Here a is one, b is minus three, c is two, and d is minus one. So sigma alpha is minus b over a: three. And sigma alpha beta is c over a: two. Now the product, alpha beta gamma. The constant term is minus one. So, is the product one, or minus one? Remember our trap! Plus one, exactly! Minus d over a: minus of minus one, over one, which is one. Many students write minus one here, and then the second answer comes out with the wrong sign. Don't do that! Now sigma alpha square: sigma alpha, the whole square, minus two sigma alpha beta. Three square is nine, minus two times two, four. Nine minus four: five. And sigma one over alpha: sigma alpha beta over alpha beta gamma, that is two over one. Just two. See, we never found a single root, and still got both answers. That is the power of Vieta!

6:30Example 2: solve x³ - 12x² + 39x - 28 = 0, roots in A.P.

Now a slightly harder one, JEE level. Solve x cube minus twelve x square plus thirty nine x minus twenty eight equals zero, given that its roots are in A P, okay? Pause the video and try it. Hint: use the topper's trick. Done? Okay, let's check. Take the roots as a minus d, a, a plus d. Their sum is three a. From Vieta, the sum is minus of minus twelve, over one: twelve. This double arrow means implies: three a is twelve, so a is four, the middle root. Now, to find d, use the product. Four minus d, times four, times four plus d. By Vieta, the product is minus of minus twenty eight, over one: plus twenty eight. Careful with that sign! Four minus d, times four plus d, is sixteen minus d square. So sixteen minus d square, times four, equals twenty eight. Divide by four: sixteen minus d square is seven. So d square is nine. Now, d can be three, or minus three. So do we get two different sets of roots? No! This plus minus sign covers both, and both give the same roots. With d equal to three: four minus three is one, then four, then seven. With d equal to minus three, the same numbers come in reverse order. So the roots are one, four and seven. And a topper's habit: always check with the relation you did not use. Pair products: one times four is four, four times seven is twenty eight, seven times one is seven. Four plus twenty eight plus seven: thirty nine, exactly the coefficient of x. Tick! Absolutely perfect.

8:24Quick revision: alternating signs, built sums, A.P. and G.P.

So, quick revision. For a cubic: the sum is minus b over a, the pairs give c over a, and the product is minus d over a. Watch that product sign. Sigma alpha square, sigma one over alpha, sigma alpha cube: build them all from these three. And for roots in A P or G P, take a minus d, a, a plus d, or a over r, a, a r. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!