Univariate polynomials and what degree names

0:00One restriction first, and it decides everything that follows. Everything here uses exactly one letter. Not two, not three. One. That is a real restriction, and plenty of useful expressions break it. But with one letter, a single number turns out to describe the whole object, and that number is what this is about. The letter itself does not matter at all. x, y, z: the same thing wearing a different name.

0:32So when the letter changes halfway through, nothing has changed except the ink. Here are five of them. Four x. x squared plus one. Two y minus five. Five y cubed plus y squared plus two y minus one. And three z plus seven. Different letters, different lengths, and all of them the same kind of object. Now sort them, by looking at the biggest power in each. Four x, two y minus five and three z plus seven all top out at power one. That is three of the five.

1:11x squared plus one tops out at two, and the long one tops out at three. So look at where that biggest power actually lives. In x squared plus five x plus one, the powers present are two, one and zero. The biggest is two. In five y cubed plus y squared minus eight, they are three, two and zero. The biggest is three. And notice what is missing from that second one. There is no plain y term in it at all. There is a gap where it would be.

1:47A gap like that is not a mistake in the writing. It is a coefficient you cannot see yet. Hold on to it. It comes back, and it comes back as the sharpest question in this whole topic. That biggest power gets a name of its own. It is the degree. The degree of a one-letter polynomial is the largest power of the letter that actually appears in it. Actually appears is doing real work in that sentence.

2:19Write zero x to the fifth, plus three x. The degree is one, not five, because the x to the fifth is not there. A term with coefficient zero is not a term. One number, readable in a second. The rest of this is about why that number is worth the trouble. Before going any further, pull the long one apart. Five y cubed plus y squared plus two y minus one.

2:50The coefficient of y cubed is five. Straightforward. The coefficient of y squared is one. Not nothing. One. An unwritten one is a convention, not an absence. The coefficient of y is two. And the constant term is minus one, not one. The sign belongs to the number itself, not to the operation sitting in front of it. Now here is why degree earns a name: whole families are named after it, and nothing else.

3:24Degree three is called cubic. Five y cubed plus y squared plus two y minus one is cubic. Degree two is quadratic. x squared plus five x plus one. Degree one is linear. Three z plus seven. Degree zero is called constant. Which raises an obvious question, and the answer to it is the best thing in this topic. What on earth is a polynomial of degree zero? Take the number eight. Just eight, sitting on its own.

4:01Is that even a polynomial? And if it is, where is its power? Here is the move. Write it as eight x to the zero. Any number other than zero, raised to the power zero, is one. So eight x to the zero and eight take the same value everywhere the first one is defined. Nothing about the number has changed. Only the way it is written. Compare that with rewriting eight as eight x. That agrees with eight at exactly one place, when x is one, and nowhere else. Eight x to the zero agrees everywhere.

4:40But now the power is visible, so the rule can read it off. Largest power zero. Degree zero. Constant. And that is the evidence that degree was the right thing to sort by. Sorting by it leaves nothing over. Now back to that gap. Four z cubed plus five z squared minus eleven. What is the coefficient of z? There is no z term written anywhere in it, so a lot of people say there is no coefficient.

5:13That is not the same statement as the coefficient is zero, and only the second one is right. Write it with the gap filled in. Four z cubed plus five z squared plus zero z minus eleven. Exactly the same polynomial. It takes the same value at every number you feed it. And now the question is not strange at all. The coefficient of z is zero. Two things degree is not.

5:45It is not the number of terms. Five y cubed plus y squared plus two y minus one has four terms and degree three. x to the fourth minus three x cubed plus six x squared minus two x plus seven has five terms and degree four. Four x has one term and degree one. And that match is a coincidence, not a pattern. Of those three, it is the only one where the two numbers agree.

6:17Degree is not the biggest coefficient either. In five y cubed plus y squared plus two y minus one the biggest coefficient is five, and the degree is three. Look up at the exponent. Not along the line at the numbers. One more trap, and it is the one that quietly costs marks. In x to the fourth minus three x cubed plus six x squared minus two x plus seven: what is the coefficient of x squared, and what is the coefficient of x cubed?

6:54Six, and minus three. In that order, because the question asked for x squared first while the expression writes x cubed first. The two orders are opposite. Read them off left to right and you hand back the right two numbers attached to the wrong two powers. And it is minus three, not three. Same rule as the constant term: the sign belongs to the coefficient. So why does this one number deserve a name of its own?

7:28Because it predicts something, and here is the thing it predicts. Take five y cubed plus y squared plus two y minus one, and work out its value at zero, then one, then two, then three, and so on. Now subtract each answer from the next one. That gives you a new list. Do the same thing to that list. And again. After three rounds, every number you have left is thirty. That is five times three factorial: the leading coefficient, times three factorial. Do a fourth round and you get nothing but zeros.

8:07So degree three is not just a description of how it is written. It is the number of rounds of subtracting it takes to go flat. Try it on a quadratic and it goes flat after two rounds. On a linear one, after a single round. Every time, the degree says in advance how many rounds it will take. One letter, and one number that sorts all of them. The degree is the largest power actually present. Zero, one, two and three give you constant, linear, quadratic and cubic.

8:45A missing power has coefficient zero. It does not have no coefficient. A sign belongs to its coefficient, and an unwritten one is still a one. And from here everything narrows to degree one. Linear. The smallest case that is interesting at all, and the one worth taking apart properly.