A zero is exactly a point where the curve meets the horizontal axis

0:00We already know what a zero is. It is an input the polynomial returns zero for, and you test one by feeding it in and looking at the answer. No picture was needed for any of that. So why turn to drawings now? Not because the arithmetic failed. Because there is a second way of saying the same thing, and the second way makes something visible that the first one hides.

0:27It tells you how many zeroes there are, and roughly where, before you have found a single one of them. And it does that by turning a question about numbers into a question about a shape. The whole of this rests on one move, which is writing y equals p of x. So let us be very careful about what that line means. Take any polynomial and any input. Feed the input in, and you get an output.

0:57Now write those two numbers down as a pair: the input first, the output second. That pair is a point. Do it for another input, and you get another point. Do it for every input there is, and the points you get are the curve. So the curve is not a picture somebody drew of the polynomial. It is a record of what the polynomial does, one entry per input.

1:24And the thing to hold on to is the second coordinate. The second coordinate of every point on that curve is the output of the polynomial at the first coordinate. That is not a property of the curve. It is the definition of the curve, and everything in this video comes out of it. Start with something we have already solved. Two x plus three. Feed in minus two: two times minus two is minus four, plus three is minus one.

1:57So the point minus two, minus one is on the curve. Feed in two: two times two is four, plus three is seven. So the point two, seven is on the curve. Feed in zero and you get three, which is the constant term, so nought, three is on it too. Now plot those. And here is a thing we are not going to prove, because it comes from earlier work and this video takes it as known.

2:26For a polynomial of degree one, those points lie on a straight line. Not roughly on one. On one. Now look at where that line meets the horizontal axis. It happens once, and it happens between the marks at minus one and minus two. The crossing is at minus three over two. And you have seen that number before. Minus three over two is exactly what came out when we solved two k plus three equals zero.

2:59That is not a happy accident and it is worth saying why, slowly. A point sits on the horizontal axis when its second coordinate is zero. The second coordinate of a point on this curve is the output of the polynomial there. So the point is on the axis exactly when the output is zero. Which is exactly when the input is a zero of the polynomial. Two sentences, one fact.

3:30Now do it without picking numbers. Take a x plus b, with a not zero. Its crossing is wherever the output is zero, and the output is zero at exactly one input, which is minus b over a. One input, so one crossing. Never two, and never none. And notice what that does not depend on. It does not depend on how steep the line is. A shallow line and a very steep line both meet the axis exactly once.

4:03Twelve different steepnesses were checked, from minus six through to six, and the number of crossings never moved off one. Steepness moves where the crossing is. It does not move how many there are. Now for something that is not a straight line. X squared minus three x minus four, which is the polynomial whose zeroes we already know. Take eight inputs, from minus two up to five, and work out the output at each.

4:36At minus two: four plus six minus four, which is six. At minus one: one plus three minus four, which is zero. At nought: minus four, the constant term. At one: one minus three minus four, minus six. At two: four minus six minus four, minus six again. At three: nine minus nine minus four, minus four. At four: sixteen minus twelve minus four, zero. At five: twenty-five minus fifteen minus four, six.

5:09Eight inputs, eight outputs, and every one of them is a point. Before we plot anything, read that row of outputs again. Six, zero, minus four, minus six, minus six, minus four, zero, six. It goes down and then it comes back up. And every value appears twice. Six at both ends, zero next to each end, minus four twice, minus six twice in the middle. The row reads the same backwards as forwards.

5:42That is a fold, and the fold is at three over two, halfway between the two middle inputs. This is not a peculiarity of this polynomial. Every quadratic folds about the point minus b over two a. It was checked by folding six hundred and forty-eight quadratics at five places each, three thousand two hundred and forty folds, and it held every single time. And to be sure that is a real property and not something that would hold for anything, the same fold was tried five hundred times on cubics.

6:20It failed on four hundred and ninety-six of them. So the symmetry is a fact about quadratics, and you can see it in the numbers before you draw a thing. Now plot the eight points. Minus two, six, up on the left. Minus one, nought, sitting on the axis. Nought, minus four, below it. One, minus six, and two, minus six, the two lowest. Three, minus four, coming back up.

6:50Four, nought, on the axis again. And five, six, up on the right. The shape is exactly what the paired outputs promised: down, across the bottom, and back up, symmetric about the middle. Now join them with a smooth curve. And that joining is the one part of this whole picture that is not exact, which we will come back to. Look at where this curve meets the horizontal axis.

7:20Twice. Once at minus one, and once at four. And those are the two numbers we found by arithmetic, with no drawing anywhere in sight. That agreement is the point of the whole topic, so do not let it go past quickly. The arithmetic route fed minus one into the polynomial and got zero out. The drawing route looked at where the curve touched down. Those two routes share nothing.

7:50One is a substitution and the other is a picture. And they land on the same two numbers, because they were never two questions. Let us state it once in general, and argue it rather than announce it. Take any polynomial p and any number k. Claim: k is a zero of p exactly when the point k, nought lies on the curve. Here is the argument, and it is three lines.

8:18The point of the curve at first coordinate k is the point k, p of k. That point has second coordinate zero exactly when p of k is zero. And p of k being zero is what it means for k to be a zero. So the two statements are not two statements. There is no theorem being proved here and nothing to be surprised by. It is one fact, written once in arithmetic and once in geometry, and the only reason it feels like news is that the two notations look so different on the page.

8:58Now a trap, and it catches people constantly. Look at where the curve meets the upright axis. That is the point nought, minus four. Is minus four a zero? No, and neither is nought. That point is the output at input nought, which is the constant term. Only the horizontal axis has second coordinate zero, so only the horizontal axis carries zeroes. The confusion is really about which of the two coordinates you are reading.

9:31Asking whether k, nought is on the curve and asking whether nought, k is on the curve are different questions. Across one thousand seven hundred and fifteen cases, those two questions gave different answers two hundred and fifty-two times. That is not a rare collision. It is one in seven. Back to the one dishonest part of the picture. We had eight points, and we joined them with a smooth curve.

10:01The eight points are exact. The joining is a guess. Nothing so far tells you the curve does not shoot off and come back between two of the plotted points. So how bad is that guess? It was measured. Among quadratics with whole coefficients up to four, take every one that meets the axis twice and sample it at whole numbers only. Not one of them hides a crossing. Every single one shows its crossings to a table of whole numbers.

10:34Then widen the coefficients to eight and try again: two thousand eight hundred and thirty-two of them cross twice, and forty of those hide both crossings between two consecutive whole numbers. So it does happen. It is rare, and it does not happen at all for the small ones, and that is why joining the dots is safe here rather than safe everywhere. One more thing the picture tells you that the arithmetic hides.

11:06A straight line always meets the axis once. A curve of degree two does not have to meet it at all. Six hundred and forty-eight quadratics were checked two independent ways: once by walking along the curve and watching which side of the axis it was on, and once by an algebraic test that never substitutes a number at all. The two routes agreed on every one of the six hundred and forty-eight.

11:36Three hundred and eighty-four of them cross the axis twice. Twenty-four touch it exactly once and turn back. And two hundred and forty never reach it at all. That last group is the interesting one, and nothing we have done rules it out. A polynomial can perfectly well have no zeroes, and its curve is what that looks like: a shape that stays entirely on one side. So what has the drawing actually bought us?

12:08Not a method for finding zeroes. We found the zeroes of x squared minus three x minus four by arithmetic, before any of this, and the drawing did not help with that. What it bought is an answer to a different question: how many are there. You can look at a curve and count crossings without solving anything. That is a genuinely useful skill, and it is the one worth practising: given a drawing, say how many zeroes and roughly where.

12:40The plotting is the slow part and it is not the valuable part. Reading is. And the reason reading works at all is the one line we argued in the middle of this video. A point is on the horizontal axis exactly when its second coordinate is zero, and its second coordinate is what the polynomial returned. Zero out, and the curve touches down. Same fact, twice.