Substituting a number into a polynomial, and what makes it a zero

0:00Here is a way of looking at a polynomial that makes everything after it much easier. It is a machine with one slot. You put a number in the slot, the machine does its arithmetic, and a number comes out. The expression on the board is not a thing sitting still. It is a set of instructions waiting for an input. Take x squared minus three x minus four. The instructions read: square whatever you are given, subtract three times it, then subtract four.

0:33Nothing there is about x in particular. The letter is just holding the place where a number is going to go. So the sensible question to ask about this machine is not what x is. It is what comes out when you feed it something. Let us feed it a two. The move is to write two everywhere the letter appears, and to do that before any arithmetic happens. Two squared, minus three times two, minus four.

1:04Keep that line on the board for a moment, because it is the line people skip. Now the arithmetic. Two squared is four. Three times two is six, and it is being subtracted. And four is being subtracted. Four minus six minus four. That is minus six. So this machine, fed a two, returns minus six. One input, one output, no ambiguity anywhere. Now feed it a zero, and watch what happens to the expression.

1:39Zero squared is zero. Three times zero is zero. Both of the first two terms carry the letter, so both of them vanish. What is left standing is minus four. And minus four was already written on the board before we started. It is the term that never had a letter in it in the first place. That is not a coincidence about this expression. Every term with a letter in it dies at input zero, and the term without one survives untouched.

2:12So the output at zero is always the constant term, and you can read it off without doing any arithmetic at all. That was checked on two thousand four hundred and one polynomials, all of them fed a zero. Every single one handed back its own constant term. This is worth a name, and the name is where the first real misreading happens. We write p of x for the machine, and p of two for what comes out when two goes in.

2:44So the line we just did says p of two equals minus six. Now, the trap. P of two is not p multiplied by two. The bracket here is not a multiplication sign, and p is not a number waiting to be multiplied by anything. P is the name of the machine. The thing in the bracket is what you fed it. And the whole symbol, p of two, is a single number: the one that came out.

3:13That is why a second letter is useful. We say p of k, where k is whatever we are feeding in at the moment, and x stays as the letter the rule is written in. One letter names the rule, the other names the input. Feed it minus one, and two sign traps arrive in the same line. Written out with brackets kept: bracket minus one, squared, minus three times bracket minus one, minus four.

3:44Minus one squared is plus one, because a negative number multiplied by itself is positive. And three times minus one is minus three, which is being subtracted, so it adds three. One plus three minus four. Zero. Now here is the line a hurried student writes instead. Minus one, minus three, minus four, giving minus eight. Both signs have been lost, because the brackets were never written down. And this is not a rare slip.

4:19Substituting a negative number and leaving its sign at the door was run on one thousand seven hundred and ten substitutions. It changed the answer on one thousand four hundred and seventy of them. The brackets are not decoration; they are the whole of the difference. Let us do one more. Feed it four. Four squared is sixteen, three times four is twelve, and four is subtracted. Sixteen minus twelve minus four.

4:52Zero again. So look at what this machine has produced so far. Fed a two, it gave minus six. Fed a zero, it gave minus four. Fed a minus one, it gave zero. Fed a four, it gave zero. Two of those four outputs are not like the others. Something different has happened at minus one and at four, and everything from here on is about that difference. Here is the definition, and it is short.

5:24A number is a zero of a polynomial when the polynomial returns zero there. That is the whole thing. Minus one is a zero of x squared minus three x minus four, because feeding in minus one gives out zero. Four is a zero, for the same reason. And now the second misreading, which is even commoner than the first. A zero of a polynomial is not the number zero.

5:54There are two slots in this sentence, and the word points at the first one. The zero is what goes in. Zero is what comes out. Minus one is a zero; four is a zero; and neither of them is zero. One word, two jobs, and telling them apart is most of the battle. Now compare two questions that look almost identical when they are written down. The first: is four a zero of this polynomial?

6:26The second: what are the zeroes of this polynomial? The first one is finished before you have properly started. Feed in four, look at the answer, and if it is zero then yes. There is nothing to search, nothing to rearrange, and no chance of not knowing when you are done. The second question is a completely different animal. Nobody has handed you a candidate. You would have to produce one from nothing, and then another, and then somehow argue that there are no more.

6:59Testing is a single line of arithmetic. Finding is a method, and a method has to be built. Keep those two apart, because everything that follows is the business of turning the first into the second. The obvious plan for finding a zero is to try numbers until one works. Try one, try two, try three, and see if anything comes out zero. It is worth seeing exactly how badly that plan does.

7:31Take every polynomial of the form a x plus b whose two coefficients are whole numbers no bigger than twelve, with a not zero. That is six hundred of them. Every one of those six hundred has a zero. But only one hundred and sixty-four of them have a zero that is a whole number. The other four hundred and thirty-six have a zero that is a fraction. So on more than two-thirds of these, trying whole numbers is not a slow method.

8:04It is not a method at all, because there is nothing whole to land on. You could sit there counting upwards for the rest of your life and never hit it. That is the situation the definition rescues you from. Watch what the definition does to a linear polynomial. Take two x plus three. We want the number k that this machine returns zero for. The definition says exactly what that means: p of k is zero.

8:36So write it out. Two k plus three equals zero. The question has stopped being a search. It is an equation, and it is a one-step equation. Take the three across: two k equals minus three. Divide by the two: k equals minus three over two. And check it, because checking here costs nothing. Two times minus three over two is minus three, and minus three plus three is zero.

9:06One line to set it up, one line to solve it, one line to check it, and no guessing anywhere. Now do the same thing without picking numbers. Take a x plus b, with a not zero. The definition gives a k plus b equals zero. Take b across: a k equals minus b. Divide by a: k equals minus b over a. Read that slowly, because the sign is the whole point.

9:37On top is the constant term with its sign flipped. Underneath is the coefficient of the letter. It is minus b over a, and it is not b over a. Across those six hundred polynomials, b over a is the right answer in only twenty-four cases, and those are exactly the ones where b is zero and the two agree by accident. On the other five hundred and seventy-six it is simply wrong.

10:08Try it on four x minus ten. Here b is minus ten, so minus b is plus ten, and the zero is ten over four, which is five over two. Check: four times five over two is ten, and ten minus ten is zero. And notice where a not zero was earning its keep. It is what let us divide by a in the last step. There is one more thing this formula is telling you, quietly.

10:38It gives one answer. Not a list, not a range, one number. So a linear polynomial cannot have two different zeroes, and here is why. Suppose two numbers both did it. Call them k and m. Then a k plus b is zero and a m plus b is zero. Subtract one from the other and the b disappears. You are left with a times k minus m equals zero.

11:06Now, a is not zero. So the only way that product can be zero is for k minus m to be zero, which means k and m were the same number all along. That was also checked by brute force. Every linear polynomial with coefficients up to six was tested against every fraction with a top and a bottom up to six. All one hundred and fifty-six of them vanish at exactly one of those fractions, and not one pair of different candidates ever both came out zero.

11:44So the linear case is closed. One zero, built entirely out of the two coefficients, found in one step. Which raises the obvious question about the polynomial we started with. Its zeroes turned out to be minus one and four. Are those two numbers hiding inside its coefficients as well? And do not reach for the guessing plan again. Take every quadratic whose three coefficients are whole numbers no bigger than four.

12:16That is six hundred and forty-eight of them. Only one hundred and seventy-six have a zero you can write as a fraction at all. The other four hundred and seventy-two do not, so no amount of trying fractions will ever find one. Whatever the method turns out to be, it cannot be a search. It has to come out of the coefficients, the way minus b over a did. That is the question everything after this is built to answer, and the answer starts with a picture.