Why degree puts a ceiling on how many zeroes there can be

0:00A quadratic bends once. One bend, two arms, and however you draw it the picture has the same shape. That single bend is what limits it to meeting the horizontal axis twice at the outside. Now let the highest power be three instead of two. A cubic is allowed to bend twice. It can climb, turn over, come back down, turn again, and climb away. Which means it can leave the axis and return to it more often than a parabola ever could.

0:34So the obvious question is how much more often. Is there any limit at all, or can a curve be made to wander across the axis as many times as you like? There is a limit, and the whole of this is about where it comes from. Start with a cubic and just do the arithmetic. Take x cubed minus four x. Feed it five inputs, from minus two up to two.

1:01At minus two, the cube is minus eight and four x is minus eight, so the two cancel and the answer is nought. At minus one, we get minus one plus four, which is three. At nought, everything vanishes and the answer is nought. At one, one minus four gives minus three. And at two, eight minus eight is nought again. Look at that column of answers. Three of those five inputs came back nought.

1:31Three of five, from a cubic, without trying. Now put those five points on a picture and join them up. The curve comes up from below on the left. It passes through the axis at minus two. It rises to a hump, turns over, and comes back down through the axis at nought. It dips to a low turn, and then climbs back through the axis at two. Three meeting points, and they are exactly the three inputs the arithmetic already handed us.

2:04And once past two the curve is climbing and never comes back. Away to the left, past minus two, it is falling and never comes back. So three is not just what we happened to draw. It is all there is. Now a cubic that does something else entirely. Just x cubed, with nothing subtracted. At two it is eight, at one it is one, at minus one it is minus one, at minus two it is minus eight.

2:36Every negative input gives a negative answer, and every positive input gives a positive one. So the only input that returns nought is nought itself. One meeting point, on a cubic. And notice what the curve does as it arrives there. It flattens out. The slope at the origin is nought, so for a moment the curve is running level with the axis. It looks like something extra is happening at that point.

3:07It is not. The curve is level there, and it goes straight through all the same, at one point and one point only. And a third cubic, to fill in the middle. Take x cubed minus x squared. Both terms carry an x squared, so pull it out. What is left is x squared, times the bracket x minus one. That factored form hands you the answers directly. A product is nought exactly when one of the things being multiplied is nought.

3:41So either x squared is nought, which happens at nought. Or x minus one is nought, which happens at one. Two meeting points, from a cubic. And they behave differently. At one the curve cuts straight through. At nought it comes down, rests on the axis, and goes back up the way it came. The factor x got used twice there, and that is what a rest looks like. Three cubics, side by side.

4:11The first met the axis three times. The second met it once. The third met it twice. So the first thing to throw away is the idea that a cubic has three zeroes. It does not. It has at most three. All three of these have the same highest power, and their counts are three, one and two. What they share is not a number. What they share is a ceiling, and only one of them reached it.

4:41The next question is why that ceiling is there at all, because so far we have only looked at pictures. Here is the move the whole argument is built on. Suppose k is a zero, so that putting k in returns nought. Then x minus k divides the polynomial exactly. Not approximately, not with something left over, but exactly, with a remainder of nothing. Watch it happen on the cubic we started with.

5:12Two was one of its zeroes, so divide by x minus two. The division goes through and the remainder is nought. Minus two was a zero too, so divide by x plus two. That goes through as well. And nought was a zero, so divide by x. That leaves nothing behind either. Try it with a number that is not a zero and the division always leaves a remainder. A zero and a factor are two ways of saying the same thing.

5:46Now do that for every zero at once. Our cubic had three of them, at minus two, at nought and at two. Each one hands over its own bracket. x plus two, and x, and x minus two. Different zeroes give different brackets, so none of them is a copy of another. Multiply the three brackets together. The result still divides the cubic exactly. In fact it is the cubic.

6:15And here is the thing to watch: each bracket has an x in it, so each one costs a degree. Three brackets multiplied together carry a highest power of three. Not two, not one, but three, because degrees add when you multiply. So having three distinct zeroes forced a degree of at least three. This is the step everything turns on, and it is the one that gets skipped. Something that divides another thing exactly cannot have a higher degree than the thing it divides.

6:51Think about why. If it divides exactly, then the polynomial is that divisor multiplied by something else. Degrees add when you multiply, so the polynomial's degree is the divisor's degree plus whatever the other part contributes. Which cannot be less than the divisor's degree on its own. So put the two halves together. If a polynomial has r different zeroes, the product of their r brackets divides it. That product has degree r.

7:25A divisor cannot outrank what it divides. Therefore the polynomial's degree is at least r. Turn that round and you have it: r is at most the degree. The count of zeroes can never exceed the highest power. Notice exactly what that argument delivers, and what it does not. It says at most. It never says exactly. The argument only ever runs from the zeroes you have to the degree you must have.

7:57It says nothing at all about zeroes you do not have. A polynomial with no zeroes breaks no rule, because there is no bracket that has to fit anywhere. Plain x cubed is about as simple as a cubic gets, and it stops at one. So do not read a ceiling as a promise. It is a wall you cannot get past, not a place you have to reach. Nothing in that argument mentioned the number three.

8:27It works the same for any highest power at all. A polynomial of degree n meets the horizontal axis at n points at the very outside. That is the general statement, and the factor argument is what backs it. It is worth saying how that was checked here, because a ceiling is easy to fake. If you count meeting points with a method that can only ever answer up to three, then watching it never say four proves nothing.

8:57The ceiling would be built into the counting. So the counting used here was handed polynomials of degree seven and degree eight, through the very same lines. It came back with seven, and it came back with eight. It could have said more than the degree. It never did. Now the skill this buys you. Six curves, no labels, no equations. Count the meeting points on each. The first comes up from below, cuts the axis once, levels off well above it, and climbs away.

9:33That levelling is not a second meeting point, because it never reaches the axis. One. The second is an arch with both ends dipping below. It comes up through the axis and goes back down through it. Two. The third cuts, peaks, falls back through, turns low, and climbs through again. Three. The fourth is a straight line, crossing once. The fifth is a bowl sitting exactly on the axis at its lowest point.

10:06That counts, and it counts once. The sixth is a wave that crosses four times. Twelve meeting points in all, and every one of them was found by looking. Six more, and these are harder. The first is a level line drawn clear of the axis. It runs alongside forever and never touches. Nought. The second sits below the axis on the left, dips, then climbs steeply and cuts once on the right.

10:37One. The third goes up through, over, down through, and up through again. Three. The fourth is a bowl whose bottom is below the axis, so both arms cut it. Two. The fifth is a wave with four cuts. And the last one is the one people get wrong. It cuts the axis on the left, rises to a peak, comes back down and rests on the axis, rises again, comes down and rests a second time, then climbs away.

11:11Two of those three contacts are rests rather than cuts, and it is very easy to read a rest as a near miss. Three, not one. Across all twelve curves there are twenty-two cuts and three rests. A rest counts. One last warning, and it matters. The argument runs from the degree to a ceiling on meeting points. It does not run the other way. You cannot look at a picture, count the crossings, and announce the degree.

11:43A curve with three meeting points might belong to a polynomial of degree three. But four, five and six can all produce exactly three as well. The extra degrees go into brackets used twice over, or into pieces that never reach the axis at all. And that level line we counted as nought could be the simplest thing there is. Counting is what a picture gives you. The degree is not.

12:13So here is what we have. Every zero hands over a bracket. Different zeroes hand over different brackets. Those brackets multiply together inside the polynomial, and each one costs a degree. Since a divisor cannot outrank what it divides, the brackets have to fit inside the degree you started with. That is the ceiling, and it is not something the pictures happened to show us. It was forced before anything was drawn.

12:45Three cubics gave us three zeroes, one zero and two. All three obeyed the ceiling. Only one of them reached it.