The three symmetric relations that hold for a cubic

0:00A quadratic has two zeroes, and there turned out to be two things you could say about them. Add them, and you get minus b over a. Multiply them, and you get c over a. Now go up one degree. A cubic has at most three zeroes, and the question is what happens to that list. It would be reasonable to guess that you get one more line, because there is one more coefficient.

0:27That guess is right about the number and wrong about the reason, and the reason is the part worth having. Ask instead how many ways there are to combine three numbers so that the answer does not care what order you took them in. You can add all three. You can multiply all three. And there is one more, which two numbers had no room for: you can pair them up.

0:54Three combinations, so three relations. Here is the cubic we will work with: two x cubed, minus five x squared, minus fourteen x, plus eight. And here are three numbers offered as its zeroes: four, minus two, and a half. Offered, not proved. Everything that follows is a statement about zeroes, so before any of it means anything, these three have to earn the name. Put four in. Two times sixty-four is a hundred and twenty-eight, minus five times sixteen is eighty, minus fourteen fours is fifty-six, plus eight.

1:36A hundred and twenty-eight minus eighty minus fifty-six plus eight is nought. Put minus two in, and you get minus sixteen, minus twenty, plus twenty-eight, plus eight, which is also nought. Put a half in: a quarter, minus five quarters, minus seven, plus eight, and that is nought as well. So all three are zeroes. And since a cubic cannot carry more than three, these are not just three of them.

2:06They are all of them. Start with the combination you already know. Add the three zeroes. Four, plus minus two, plus a half. Four minus two is two, and two and a half is five halves. Now read the polynomial instead. a is the two in front of the x cubed, and b is the minus five in front of the x squared. Minus b over a is five over two.

2:34Five halves again. So the first relation survives the change of degree completely unaltered: the zeroes still add to minus b over a, and b is still the coefficient one place down from the top. That is worth noticing before anything new arrives, because it means we are extending a pattern rather than starting a fresh list. Now multiply all three together. Four times minus two is minus eight, and minus eight times a half is minus four.

3:11Look for that in the coefficients. The constant term, d, is eight, and a is two, so d over a is four. But the zeroes gave us minus four. The sizes match and the signs do not, so the relation is not d over a. It is minus d over a, which is minus four, and now it agrees. And that is a real difference from the quadratic, where the product of the zeroes was c over a with no minus sign in front of it at all.

3:43Same idea, opposite sign, one degree apart. Nobody should have to remember which is which, and in a few minutes nobody will have to, because the expansion will say exactly where that minus sign comes from. Now the one that is genuinely new. With two zeroes there was nothing between adding them and multiplying them. With three there is a middle step: take them two at a time. Here are the three zeroes as three dots, and here are the pairs.

4:17Four with minus two. Minus two with a half. A half with four. Three dots, three connecting lines, three pairs — and that is all of them, because choosing two out of three is the same as choosing which one to leave out. Multiply each pair. Four times minus two is minus eight. Minus two times a half is minus one. A half times four is two. Now add those three products: minus eight, minus one, plus two is minus seven.

4:52And c over a is minus fourteen over two, which is minus seven. Notice what that phrase does not mean. It is not the first two multiplied together. It is every pair, and their products added. Three numbers checked is not three relations proved, so let us do what we did for the quadratic and build a cubic out of its zeroes. Call them alpha, beta and gamma. The bracket x minus alpha is nought at alpha, and the same for the other two, so the product of the three brackets is nought at all three.

5:30Multiply it out slowly, because the whole answer is in the bookkeeping. Every term is made by walking through the three brackets and taking either the x or the zero from each one. Take x from all three, and you get x cubed. Take the zero from exactly one bracket and x from the other two, and you get minus alpha x squared, minus beta x squared, minus gamma x squared.

5:59Take the zero from two of them, and you get plus alpha beta x, plus beta gamma x, plus gamma alpha x. Take the zero from all three, and you get minus alpha beta gamma. That is every possible way through, so that is the whole expansion. Now collect it. x cubed, minus the bracket alpha plus beta plus gamma times x squared, plus the bracket alpha beta plus beta gamma plus gamma alpha times x, minus alpha beta gamma.

6:36Look at what has appeared as the coefficients. The x squared coefficient is the sum of the zeroes, with a minus. The x coefficient is the zeroes two at a time, with a plus. The constant is the product of all three, with a minus. Those are our three combinations, and nothing else survived. Now multiply the whole thing by a, exactly as we did for the quadratic, because scaling by any number that is not nought moves no zero.

7:09Set that beside a x cubed plus b x squared plus c x plus d and match the coefficients one at a time. b is minus a times the sum. c is a times the pairs. d is minus a times the product. Divide each by a and the three relations fall out. So look again at where those minus signs came from. Each bracket is x minus something, so every time a term takes the zero rather than the x, it picks up one minus sign.

7:45A term built from one zero carries one minus sign, so it is negative. A term built from two zeroes carries two, and two minus signs cancel, so it is positive. A term built from three carries three, so it is negative again. Minus, plus, minus. That is not a rule to memorise; it is just counting how many brackets gave up their minus sign. And it tells you what happens next without doing any work.

8:16At degree four the pattern continues: minus, plus, minus, plus. It also explains the thing that looked arbitrary earlier. For a quadratic the product of the zeroes used two zeroes, so it came out positive, as c over a. For a cubic the product uses three, so it comes out negative, as minus d over a. Not two unrelated formulas at all — the same rule, counted at different lengths.

8:49So here they are, for any cubic a x cubed plus b x squared plus c x plus d, with a not nought. The zeroes taken one at a time — added — give minus b over a. The zeroes taken two at a time, every pair, added, give c over a. The zeroes taken all three at once give minus d over a. One line for each way of combining three things, the signs alternating because of where the brackets put them, and every one a ratio to the leading coefficient because that is the number that fixes the scale.

9:30One more, worked the same way round, because the order is the lesson. Take three x cubed, minus five x squared, minus eleven x, minus three, with three, minus one, and minus a third offered as its zeroes. Substitutions first. At three: eighty-one, minus forty-five, minus thirty-three, minus three, which is nought. At minus one: minus three, minus five, plus eleven, minus three, also nought. At minus a third the arithmetic is worth doing carefully, because everything hangs on that minus.

10:11Over a common denominator of nine it is minus one, minus five, plus thirty-three, minus twenty-seven, all over nine. Minus one minus five is minus six, plus thirty-three is twenty-seven, minus twenty-seven is nought. So the third zero really is minus a third. If you had read it as plus a third, both the middle relation and the product would come out wrong, and the substitution is what catches that.

10:44Now, and only now, the relations. Add them: three, minus one, minus a third, which is two minus a third, or five thirds. And minus b over a is five over three. Two at a time: three times minus one is minus three; minus one times minus a third is plus a third; minus a third times three is minus one. Minus three, plus a third, minus one is minus eleven thirds.

11:16And c over a is minus eleven over three. All three multiplied: three times minus one is minus three, times minus a third is one. And minus d over a is minus, minus three, over three, which is one. Three for three. One last thing about the number three, because it is easy to get the reason wrong. A cubic has four coefficients: a, b, c and d. So why are there only three relations?

11:49Because the relations are not matched to coefficients. They are matched to ways of combining the zeroes, and there are three of those. The fourth coefficient, a, is not left over — it is doing a different job. It sets the scale, and every relation is a ratio to it. That is why a appears in the denominator of all three rather than getting a line of its own. There is one thing about all this that is easy to check badly, and it is worth saying how it was checked here.

12:24The middle relation counts pairs, and three zeroes make three pairs. But a piece of machinery that can only ever make three pairs gives the right answer for every cubic in existence, so testing it on cubics proves nothing at all. The pairing used here was therefore handed four zeroes, where it must come back with six pairs, and five zeroes, where it must come back with ten. It did.

12:53The same went for the alternating signs: rather than writing them down, the checking recorded which sign each relation actually needed, on every polynomial, at four different degrees, and the pattern of minus and plus came back out of the measurements. Two thousand one hundred and eighty-four cubics were built from chosen zeroes and every one had those zeroes confirmed by substitution before anything else was asked. And the zeroes were then found again from the coefficients alone by something using no formula at all — hunting for sign changes and squeezing each zero into a bracket narrower than a millionth — so the three relations could be checked against numbers that had never seen b, c or d.

13:48So: three relations, because there are three ways to combine three zeroes symmetrically. One at a time gives minus b over a, two at a time gives c over a, and all three at once gives minus d over a. The signs alternate because each bracket is x minus something, so a term built from an odd number of zeroes ends up negative. Two at a time means every pair, added — not the first two multiplied.

14:20And none of it can be checked on numbers that have not first been shown to be zeroes, which is why both cubics here were substituted into before they were combined. Going up a degree did not introduce a new idea. It introduced one more way of combining what was already there.