A polynomial as an input–output machine

0:00Up to now, every question about a polynomial has been the same question. What is it? What is its degree, what are its coefficients, what is its constant term. Change the question. What does it do? Not what it is made of. What it does to a number. Two x plus three does something. Hand it a number, and it hands one back. That is worth drawing, because once you draw it the rule and the values come apart. And every remaining question in this topic is about one or the other.

0:38Feed it four. Two times x plus three, with x set to four. Multiply first. Two times four is eight. Then add. Eight plus three is eleven. Four goes in, eleven comes out. And notice the letter did not get erased. Substituting is choosing a value for x, not deleting it. The rule is still sitting there, waiting for the next number. And watch what happens if you add before you multiply. Four plus three is seven, doubled is fourteen. Fourteen is not the answer. The order is not decoration.

1:20Now feed it minus six. Multiply first. Two times minus six is minus twelve. Then add. Minus twelve plus three is minus nine. Here is where people lose it. They work out that two sixes are twelve, then put a minus in front of the whole of twelve plus three, and land on minus fifteen. Only the two-x term went negative. The plus three was never touched. And there is a second wrong route, which is worse, because it gets the right answer. Put the minus in front of twelve minus three instead, and out comes minus nine.

2:01Right number. Wrong reason. And you cannot catch it by trying more negative inputs, because that route agrees with the machine at every negative number there is. Feed it four and it says minus five, where the machine says eleven. One more input, and it is the easiest one there is. Feed in zero. Two times zero is nothing at all, so what comes out is just the three. That is the constant term, standing on its own.

2:34Which gives the constant term a meaning you can point at. It is what the machine returns when nothing goes in. And none of this depends on the letter. Five x minus three and five s minus three are the same machine with a different label on the slot. Run that one through. Five x minus three at zero gives minus three. At minus one, minus eight. At two, seven.

3:03And here is one that catches almost everybody. Seven s squared minus four s plus six, at s equals minus three. Minus three squared is nine, and seven nines are sixty-three. Then minus four times minus three is plus twelve. Both of those came out positive. Sixty-three plus twelve plus six is eighty-one. So here is the machine, drawn properly. Three parts, doing three different jobs. A card goes in at the top, carrying the input. x equals four.

3:43A panel on the front shows the rule. That panel never changes. A tag comes out of the side, carrying the output. Eleven. Change the card and the tag changes with it. The panel does not move. That separation is the whole reason for drawing this. Put in two and the tag reads seven. Put in three and it reads nine. Same machine every time. Nothing about it shifted. The panel needs a way of naming what it produces, so it gets a letter of its own.

4:19y equals two x plus three. y is not a second unknown. It is a name for whatever comes out. So the card going in says x equals four, and the tag coming out says y equals eleven. So there are two letters on the machine now, and they have different jobs. x names what you choose. y names what you get. That second letter is going to matter later, when these pairs get plotted. For now it is simply the label on the display.

4:54Now the property that makes this a machine rather than a lottery. Line the inputs up on the left, the outputs on the right, and draw one arrow from each. One, two, three. Out come five, seven, nine. Every input has exactly one arrow leaving it. No input ever splits into two answers. Put the same number in twice, and you get the same thing out twice. An arrow that split would mean the machine had hesitated. That the same number could come back as two different things.

5:29That sounds too obvious to bother saying. It is not. It is exactly the thing being claimed. A rule that commits to exactly one output for each input has a name. It is called a function. Two x plus three is a function. And at this stage, that is the entire content of the word. Nothing more than that. It is a promise about behaviour, not a new piece of notation.

5:59And a function is not the same thing as a formula. Two rules that look completely different can be one function, so long as they agree at every input. x plus the quantity x plus three is the same function as two x plus three. Meanwhile a rule that could hand back either of two answers is not a function at all, however tidy it looks written down. There is a great deal more to say about functions, and it gets said in later years. This is the part that is needed now.

6:37Now swap the panel. Keep the machine exactly as it is, and change the rule to ten x minus x squared. That one is the area of a rectangle bent out of a twenty centimetre wire, with one side called x. Feed in six. Ten sixes are sixty. Six squared is thirty-six. Sixty take away thirty-six is twenty-four. Six goes in, twenty-four comes out. And look at what the labels say. Centimetres went in. Square centimetres came out. The machine changed the kind of quantity it was holding.

7:18Now try four. Ten fours are forty. Four squared is sixteen. Forty take away sixteen is twenty-four. Twenty-four again. Two different inputs, and the same output. Which is not a fluke of the arithmetic. Six by four and four by six are the same rectangle, turned round. So this machine's output does not tell you what went in. Twenty-four could have come from four or from six, and the tag does not say which.

7:53Two x plus three can never do that. Take any two different inputs. The gap between the outputs is two times the gap between the inputs. Two times something that is not zero is not zero. So the outputs cannot be equal. Different in, different out. Every time. And that is not a property of polynomials in general. The quadratic just broke it, and a rule that is simply the number seven breaks it as hard as anything can, by returning seven for everything.

8:29It belongs to degree one, with a coefficient that is not zero. So the machine is drawn, and it buys three questions. Feed it something and ask what comes out. That is substitution, and it is the one we have been doing. Name an output instead, and ask which input produces it. That is an equation. Or keep every input-and-output pair at once and look at the whole collection together. That is a graph, and that is next.

9:02One machine. Three questions. What separates them is only which end of it you are holding.