Degree as the label that separates linear, quadratic and cubic

0:00Every polynomial carries a number that decides what it is allowed to do. That number is its degree. It sounds like a filing label. Linear, quadratic, cubic — three boxes to sort things into. It is not a label. It is a prediction. It decides, before you have solved anything, how many times the curve is allowed to cross the axis. And here is the claim that settles it. You can find the degree of a polynomial without ever looking at the polynomial.

0:36First the rule, which is short. The degree is the largest power of the variable that actually appears. Take 4x plus 2. The largest power is one. Degree 1. Take 2y squared minus 3y plus 4. Degree 2. Take 5x cubed minus 4x squared plus x minus root 2. Degree 3. Notice the letter changed and nothing else did. The letter carries no meaning. It is a place to put a number.

1:10Now a harder one. 7u to the sixth, minus three halves u to the fourth, plus 4u squared, plus u, minus 8. The powers there are nought, one, two, four and six. The largest is six, so the degree is 6. And look at what is missing. There is no cube and no fifth power. The rule asks for the largest power present, and says nothing whatever about the ones that are absent.

1:40Now the promise from the start. The degree without looking. Here is a polynomial. I am not going to show it to you. All you get is what it does: its value at nought, at one, at two, at three, and so on. Minus 2, minus 1, 6, 25, 62, 123. Subtract each number from the one after it. 1, 7, 19, 37, 61. No pattern yet. Do it again.

2:116, 12, 18, 24. Those are climbing evenly. Do it once more. 6, 6, 6. Constant. And one more time: 0, 0. It took three rounds of subtracting to flatten it to nothing. Three rounds. Degree 3. That happens every time, for every polynomial there is. The polynomial was x cubed minus 2, and the difference table found the 3 without being told a single thing about powers. The degree is not something written on the expression. It is something the expression does.

2:51So there are two completely different ways to find a degree. Read the largest power off the expression. Or difference its values until they die. There is a third: fit a curve through the points and see how much curve you need. Every polynomial with powers up to the fourth and small whole coefficients was put to all three. That is 3124 expressions. The reading route and the differencing route agreed on every single one, and so did the fitting route.

3:26Two of those three never see the expression at all. Now three ways of getting this wrong, worth more than the rule itself. The first: degree counts the terms. 3x cubed minus 2x squared plus x minus 1 has four terms, and degree 3. Root 2 times x cubed has one term. Also degree 3. One term, four terms, same degree. In the sweep, expressions of degree 3 turned up with as few as one term and as many as four, and so did expressions of degree 4.

4:04Count powers. Never count terms. The second mistake: the first term tells you the degree. Look at 2x plus 5 minus x squared. It opens exactly like a linear expression. 2x. Same as 2x minus 3. And it is not linear, because the square is sitting at the far end. Scan the whole thing before you decide. Because writing the terms in a different order is not a different expression.

4:36Every specimen in this video was rewritten in every possible order of its terms — 230 rewritings in all. The degree came out the same in every one of them. There is a stronger version of the same point. Write extra terms in front with zero as their coefficient. 372 expressions were rewritten that way, with zeroes stacked in front of them. Not one degree moved. A power written with a zero in front of it is not there.

5:10The third mistake is the biggest, and it is about expressions that are refused. One over x minus 1. Root x plus 2. And one over x squared plus 2x plus 3. None is a polynomial, and the usual explanation is that the power is not a whole number. That explanation is true of exactly one of the three. Root x is x to the power a half, so that one is fair.

5:40But look at the last one. Every power of x in sight is whole. And it is defined for every real number you can name. Its bottom line never reaches zero, anywhere. So it is not disqualified for a broken power, and not for blowing up. The reason that catches all three is about degree. When you multiply two polynomials, their degrees add. That was checked on 624 products, and it held on every one.

6:13So suppose one over x minus 1 were a polynomial, p. Then p times x minus 1 would equal 1. The left side has degree at least one. The right side has degree nought. 624 candidates were multiplied out. Not one gave 1. Root x plus 2 dies the same way. If it were a polynomial, squaring it would give x. But squaring doubles the degree, and doubling a whole number never lands on one.

6:46624 candidates were squared. Not one gave x. Each of those expressions would need a degree, and there is no whole number available to be it. So: degree 1. The linear family. Here are six of them, and the point is how little they look alike. 2x minus 3. Root 3 times x, plus 5. y plus root 2. x minus two elevenths. 3z plus 4. Two thirds of u, plus 1.

7:20Four different letters between them, and four of the six carry a coefficient that is not a whole number. Coefficients are allowed to be any real number. That is not a concession, it is the definition. What every one of them shares is the only thing that counts: the largest power is one. Degree 2. The quadratic family. The word is worth a moment. It comes from a word meaning square, not from anything meaning two.

7:52The family is named after the geometry of the second power, and not after the number 2 at all. Here are six quadratics. 2x squared plus 3x minus two fifths. y squared minus 2. 2 minus x squared plus root 3 x. A third of u, minus 2u squared, plus 5. Root 5 times v squared, minus two thirds of v. And 4z squared plus a seventh. Only one of those six is written the tidy way, with all three powers in order, highest first.

8:31Two do not even lead with the square, and two have no first-power term at all. Every one is quadratic. The shape on the page is not the mathematics. So what is the general quadratic for? It is written a x squared, plus b x, plus c. a, b and c are real numbers, and a is not zero. Read it letter by letter. a multiplies the square, b multiplies x on its own, and c is the constant term.

9:03There is a way to see that c is the constant without looking at it: put nought in for x. Every term with an x in it vanishes, and what is left standing is c. That was checked across the whole sweep. Then every choice of a, b and c with a non-zero was tried — a hundred of them — and every one came out quadratic. So the general form is not a shape you have to write things in. It is a claim that every quadratic can be written that way.

9:38Degree 3. The cubic family, and the general form follows the same pattern. a x cubed, plus b x squared, plus c x, plus d. Four letters now, and a still not zero. Five hundred choices with a non-zero were tried, and every one is cubic. But look at how little a cubic has to look like that. 2 minus x cubed. Just x cubed, on its own. Root 2 times x cubed.

10:08A one-term cubic is a perfectly ordinary cubic. Here is 3 minus x squared plus x cubed. Three terms, the cube written last, and no plain x anywhere. It is missing a term and has the others in the wrong order, and it is cubic all the same. Only one of the five is written out in full. Now the condition that looks like small print. a is not zero.

10:37It is not small print. It is the entire meaning of the word quadratic. Set a to nought in a x squared plus b x plus c, and watch. The square disappears. What is left is b x plus c. Every pair of small whole numbers b and c was tried — 121 of them. 110 of those dropped to degree 1. 10 dropped all the way to degree nought, a lone constant.

11:07And one, where b and c were both nought, was left with nothing at all. Not one of the 121 was still quadratic. Set a to nought in the cubic form, across 343 cases, and not one is still cubic. So a is not zero is not a technicality bolted on at the end. It is the requirement that the square really be there. So what is all this labelling actually worth?

11:38This. The degree is a ceiling on how many times the curve can meet the horizontal axis. Degree 1, at most one crossing. Degree 2, at most two. Degree 3, at most three. That is a strong claim, so it was measured. For all 3124 expressions in the sweep, the real zeroes were counted exactly. Not one ever had more zeroes than its degree. And the ceiling is not generous: at every degree, something in the sweep reaches it.

12:13Look at the hundred quadratics on their own. 56 of them cross the axis twice. 8 just touch it, once. And 36 never meet it at all — no real zero anywhere. Nought, one or two. Never three. The cubics behave differently. All 500 of them have at least one real zero. An odd degree cannot avoid the axis. An even one can. That is what the label buys you: before you have solved anything or drawn anything, the degree has already told you what is possible.

12:52So, the whole thing. The degree is the largest power actually present. Not the first term you see, not the number of terms, and not affected by the order they are written in. Degree 1 is linear, 2 is quadratic, 3 is cubic, and the general forms are claims about those families rather than costumes they have to wear. And an expression with the variable under a division line or inside a root has no degree to give at all.

13:25But the sentence to keep is the one from the difference table. The degree is not a name somebody wrote on the expression. It is a fact about how the expression behaves — and you can measure it from the outside, without ever reading a single power.