What makes a polynomial linear, and the equation you get by fixing its value

0:00Degree one has a name of its own. Linear. A name on its own is not worth much. This one buys something, and the something is watchable. Here is the claim. Take a polynomial of degree one. Feed it inputs that are evenly spaced, at any spacing you like, and the answers come out evenly spaced too. Equal steps in, equal steps out. Every time, and it does not matter where you start.

0:31And nothing of higher degree manages it. That is what makes this a fact about degree one, rather than a fact about one lucky example. So let us watch that happen, and then work out exactly how big the step is, because that turns out to be the interesting part. Start with the simplest linear polynomial in sight. A square, with a side of x. Its perimeter is the distance all the way round, so add up the four sides. x plus x plus x plus x.

1:07Four x. One term, degree one, and the coefficient is four. Notice that the four x came out of counting sides. It is the conclusion, not the starting point. Now grow that side, and watch what the perimeter does. Sides of one, one and a half, two, two and a half, and three centimetres. The perimeters are four, six, eight, ten and twelve. Look at the gaps between them. Four to six is two. Six to eight is two. Eight to ten, two. Ten to twelve, two.

1:42Four equal steps out, from a side that grew in four equal steps of half a centimetre. That is the claim, working. And it holds wherever you start. Begin at five centimetres instead, still stepping by a half, and the perimeters still climb by two. But there is something odd about that two, and it is worth stopping for. The coefficient is four. The side went up by a half each time. And the perimeter went up by two.

2:15Two is not four. So the step in the answer is not the coefficient. Here is what it actually is. Four times a half is two. The output step is the coefficient multiplied by the input step. Test that the other way round. Step the side by a whole centimetre instead, and the perimeters run four, eight, twelve, sixteen. Steps of four. Same polynomial. Different input step. Different output step, and both times it came to the coefficient times whatever you chose.

2:52The step size is fixed, but only once you have fixed how you are stepping. A second one, and this time not a shape. A club charges two hundred to join, once, and then fifty for every match you play. Play one match and you have handed over two hundred and fifty. Two matches, three hundred. Three, three hundred and fifty. Then four hundred, then four hundred and fifty. After m matches you have paid two hundred plus fifty m.

3:26Degree one again. Linear again. And the totals climb by fifty each time. Here the input is stepping by one, so the output step and the coefficient happen to land on the same number. Now look at what that run of totals does not contain. Two hundred and fifty, three hundred, three hundred and fifty, four hundred, four hundred and fifty. Two hundred is not in there. It never appears at all.

3:58And yet it is the joining fee, which is arguably the most important number in the whole arrangement. Put m equal to zero, meaning nobody has played anything, and the polynomial hands you two hundred exactly. The constant term is the value before the process starts. It is very often the one number your data never shows you. And here is the trap that follows. The totals climb by fifty, so it is tempting to say the polynomial is simply fifty m.

4:33It is not. Fifty m would give you fifty, a hundred, a hundred and fifty. The real totals are two fifty, three hundred, three fifty. The constant did not change the steps. It lifted the entire run. So what happens if the degree is not one? Take seven s squared, minus four s, plus six, and feed it zero, one, two, three, four, five. You get six, nine, twenty-six, fifty-seven, one hundred and two, and one hundred and sixty-one.

5:09The steps are three, seventeen, thirty-one, forty-five, fifty-nine. Not equal. Growing. Though look what they grow by. Fourteen, every time. The steps of the steps are equal, which is degree two going flat exactly one round later than degree one did. And notice the procedure never changed. You put a number in and worked it out. What changed is the pattern in the answers, which is the thing degree was telling you about all along.

5:44Everything so far has asked the same question. What comes out? Now turn it around. And notice that up to this point there has been nothing to solve. Two hundred plus fifty m is not a question. It is a different number for every m you feed it. Suppose somebody has paid seven hundred and fifty. How many matches did they play? You are no longer asking what comes out. You are stating what came out, and asking what went in.

6:16Write that down. Two hundred plus fifty m equals seven hundred and fifty. And that is fifty m equals five hundred and fifty, so m is eleven. Notice what happened to the object. Pinning the value to one chosen number turned it into an equation, and an equation has an answer, which an expression never does. Here is the same move again, starting from a sentence this time. Two numbers add up to sixty-four. One of them is ten bigger than the other.

6:52Nothing there is algebra yet. The algebra begins the moment you name something. Call the smaller one x. Then the larger one is x plus ten, and you do not get to choose that. The sentence chose it for you. Their sum is x, plus x plus ten, which collects to two x plus ten. And the sentence says that sum is sixty-four. Two x plus ten equals sixty-four. Now solve it, and do not skip the middle.

7:25Two x plus ten equals sixty-four. Take ten off both sides. Two x equals fifty-four. Halve both sides. x equals twenty-seven. There are three different equations on that board, not one. Each step threw the old one away and wrote a new one. What makes a step legal is that the new equation has exactly the same answer as the old one. That is the only thing being kept. Twenty-seven. So the two numbers are twenty-seven and thirty-seven.

8:01Do not check that against your algebra. Check it against the sentence you started from. Twenty-seven plus thirty-seven is sixty-four. That is the first condition, satisfied. Thirty-seven take away twenty-seven is ten. That is the second one. And you needed both. Thirty and thirty-four also add to sixty-four. They pass the first test and fail the second. A pair that satisfies half the wording is not an answer. It is a near miss wearing the costume of one.

8:38So: degree one, doing two jobs. As a polynomial it is a promise about steps. Even spacing in, even spacing out, and the size of the output step is the coefficient times the input step. Its constant term is where the process starts, whether or not your table ever puts it on show. And the instant you fix its value, it stops answering what comes out and starts asking what went in. That is an equation.

9:10Same object. Different question. Knowing which one you are holding is most of the skill.