PrepShorts · Study sheet · Class 10 Mathematics · Chapter 2, Polynomials
Chapter 2 · Polynomials
Which way a parabola opens, and the three ways it can meet that axis
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x squared minus x plus one never crosses the horizontal axis, and that is not a failed calculation. Completed, it becomes a square plus three quarters — never negative, so never zero.
The idea
Two things fix everything a parabola can do about the horizontal axis: which way it opens, decided by the sign of the number multiplying the square, and how the turn of the curve sits relative to that axis. Take it that the curve turns just once and is balanced about that turn — both are true of every parabola, and both are not in the book, since neither is stated anywhere in this chapter, which prints no word for the symmetry and none for the turn. Granting them, the axis can be met twice, met once, or missed entirely — three outcomes and no fourth, whichever way the curve opens. So "this quadratic has no zero" is not a failed calculation. It is the third case, it is drawn in the chapter, and it is the reason the chapter says at most two rather than two.
What you should be able to do
- Predict from the sign of the leading number whether the curve of a quadratic opens upward or downward, and give the reason
- Name the three possible relationships between such a curve and the horizontal axis, and sketch each for both openings
- Read the number of zeroes off a supplied quadratic curve in each of the three cases
- Explain why a curve that touches the axis without crossing is counted as two equal zeroes, while allowing that the chapter itself also calls that case one zero — and say why both descriptions are defensible
- Argue why no fourth case is possible
- State the ceiling this argument establishes for a degree-2 polynomial, and say why it is a ceiling and not a count
- Recognise that a quadratic with no crossing is a legitimate polynomial with legitimate coefficients, not an error
Words to know
| Term | Definition in one line | First introduced |
|---|---|---|
| parabola | the curve traced by a degree-2 polynomial | §2.2, p. 13 |
| distinct | said of two zeroes that are different numbers | §2.2, pp. 14–15 |
| coincident | said of the two points of Case (i) once they have run together | §2.2, p. 14 |
| x-axis | the horizontal reference line the curve is being compared against | §2.2, p. 12 |
| zero of a polynomial | an input at which the polynomial returns 0 | §2.1, p. 11 |
| real number | the only kind of number in play here, which is what "no zero" is relative to | §2.1, p. 10 |
| opens upward / opens downward | the two directions the curve of a quadratic can take | an added phrasing; the chapter draws the two shapes and describes them in running prose without a fixed compound term |
| turning point | the single place where the curve stops falling and starts rising, or the reverse | an added term; this chapter never prints it, and nothing in Chapter 2 depends on locating it |
| touch | the explanation's shorthand for meeting the axis without passing through it | an added shorthand, not printed in this chapter; the chapter describes the same situation as two points that have come together |
Where people slip up
- "Every quadratic has two zeroes." The chapter draws two whole figures, Fig. 2.4 and Fig. 2.5, whose entire purpose is to refute this. Two is the maximum, not the outcome.
- "No zero means I made an arithmetic mistake." It means the curve missed the axis. Case (iii) is a picture of a perfectly ordinary quadratic — every coefficient real, a ≠ 0 — that simply has no real input returning 0.
- "A curve that only touches the axis has no zero, because it never crosses." The chapter counts it: the x-coordinate of the touching point is "the only zero" in that case, the place where the two crossing points have come together, which is why the book also calls it "two equal zeroes (i.e., one zero)". Both names are the book's own, and both describe one value.
- "a > 0 means the curve is above the axis." It fixes the direction of opening, and nothing else. Fig. 2.3 panel (ii) and Fig. 2.5 panel (i) both open upward and sit differently against the axis.
- "Whether there are zeroes depends on how far you extend the drawing." For an upward-opening curve the two arms rise forever, so extending the picture cannot bring them back down to the axis. Say this once; it is what makes the three cases exhaustive rather than provisional.
- "There should be a fourth case where the curve runs along the axis." That would need every input to return 0, which no quadratic with a ≠ 0 does.
- "Case (iii) means the polynomial has zeroes we are not allowed to talk about." This chapter works with real numbers throughout, and what it claims is that no real input returns 0. Leave it there — anything further is outside the book.
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Worked answers: Exercise 2.1 · Exercise 2.2
Transcript1,907 words
Every quadratic draws the same kind of curve. Not the same curve, but the same kind: one bend, and two arms going off in the same direction as each other. And there are only two ways that can point. The arms can both go up, with the bend at the bottom. Or they can both go down, with the bend at the top. That is it. There is no third shape, and nothing in between the two.
So before you know anything else about a quadratic, you can know which of those two pictures you are looking at. And you can know it from a single number. Take a x squared plus b x plus c, with a not zero. The claim is that the sign of a decides it, on its own. Positive opens upward, negative opens downward, and the other two coefficients have no say.
Here is why. Go a long way out from the origin, in either direction. Squaring a big number gives a much bigger number, and it gives a positive one whether you started positive or negative. So out there, a x squared is enormous, and it carries whatever sign a has. Meanwhile b x has only grown in proportion, and c has not grown at all. The square term takes over.
That was checked on every quadratic with whole coefficients no bigger than four, six hundred and forty-eight of them. Each was pushed outwards from the origin until its sign settled, and every single one settled on the side its leading number names. The furthest any of them had to be pushed was five. One more thing before the three cases, and it is the thing that makes there be exactly three.
The curve bends once. Once, not twice. It comes in, it turns, and after that it only ever goes one way. That was checked too: all six hundred and forty-eight turn exactly once. And to be sure the check could see a second turn, five hundred cubics were pushed through the very same comparison, and three hundred of them turned twice. So one turn is a fact about quadratics, not something the test was unable to contradict.
For an upward-opening curve that means the arms only climb after the turn, forever. Three hundred and twenty-four upward-opening quadratics were walked outwards from their own turning point, and every one of them only ever climbed. Hold on to that, because it is what stops the answer from depending on how far you extend the drawing. Now put such a curve next to the horizontal axis and ask where they meet.
First case. The bend sits below the axis and the arms go up. Coming down from the left, the curve must cross the axis to get below it. Going up on the right, it must cross again to get back above. Two crossings, at two separate places. Turn the whole picture over and the same thing happens the other way up: the bend above the axis, the arms going down, one crossing on the way down and one on the way back.
Two crossings again. So this case happens for both openings, and it looks the same in both. And we already know what those two places are. A point sits on the horizontal axis exactly when the polynomial returns zero there. So each crossing is a zero, and two crossings are two zeroes. Take x squared minus three x minus four. Its leading number is one, which is positive, so it opens upward.
It crosses the axis at minus one and at four, which are exactly the two numbers that came out of the arithmetic. Two crossings, two zeroes, and no new fact needed to connect them. That is the first case, and it is the one people expect. The whole point of what follows is that it is not the only one. Second case, and the way to see it is to slide the first one.
Start with the curve cutting the axis at two separate points, and lift the whole curve slowly upwards. The part below the axis gets shallower. The two crossings walk towards each other. And at one particular height they arrive at the same place. The curve now rests on the axis instead of dipping through it. One point of contact, not two. Lift it any further and the contact is lost altogether, but stop exactly there and you have the second case.
It happens for a downward-opening curve too, hanging from the axis by its highest point rather than resting on it by its lowest. One meeting place, either way up. Now, how many zeroes does that curve have? There is one point on the axis, so it is tempting to say one. And as a count of different numbers, one is right. But watch what the sliding did. Two zeroes travelled towards each other and arrived together.
Neither of them left. They are still both there; they have simply become the same number. And you can see that in the polynomial itself rather than in the picture. Every quadratic in the sweep that touches the axis - twenty-four of them - turns out to be its leading number times a bracket multiplied by itself. Not something like it. Exactly that, every time, all twenty-four. So the repeated factor is written into the polynomial, and calling it two equal zeroes is reading what is there.
Counting it as one distinct value is fine. Counting it as proof that a quadratic can carry an odd number of zeroes is not. Third case. Keep lifting. The bend is now above the axis, the arms go up from there, and the curve never comes down again. It misses the axis entirely. No crossing, no touch, nothing. And upside down, the same: a curve whose highest point is below the axis and whose arms go down from there.
Wholly on one side, and staying there. This is the case that makes people uncomfortable, so let us be clear about what it is. A quadratic whose curve misses the axis is not a mistake. It is not an arithmetic slip and it is not a broken polynomial. Its coefficients are ordinary numbers, its leading number is not zero, and it is a perfectly good quadratic in every respect. It simply has no input that makes it return zero.
That is an answer to the question, not a failure to answer it. And it is not rare. Of those six hundred and forty-eight quadratics, two hundred and forty never reach the axis at all. That is more than one in three. So no zeroes is not an edge case you can hope not to meet. Here is one you can check by hand. X squared minus x plus one.
Rewrite it as x minus a half, all squared, plus three quarters. Multiply that out and you get back exactly what you started with. Now the left-hand piece is something squared, so it is never negative. The smallest it can be is nought, and that happens at x equal to a half. So the whole thing is never less than three quarters. Three quarters is above zero, so the curve never reaches the axis.
No square roots, no formula, and nothing taken on trust. Here is a second one, x squared plus root five. Root five is a positive number a bit over two, and x squared is never negative, so the total is never less than root five. Which is comfortably above the axis, everywhere. Both of those are third-case curves with an equation attached. So there are three cases, and each of them happens for both openings.
That is six pictures, and it is worth seeing them as a grid. Three cases across, two openings down. Across the six hundred and forty-eight, the grid comes out even. A hundred and twenty upward-opening curves miss the axis, and a hundred and twenty downward-opening ones do. Twelve touch it opening upward, twelve opening downward. A hundred and ninety-two cross it each way. The opening and the case are independent of each other, and that is the point of drawing all six.
Which kills a tempting shortcut. A positive leading number does not mean the curve sits above the axis. It fixes which way the arms point, and nothing else. Of the three hundred and twenty-four upward-opening quadratics here, two hundred and four touch or cross the axis. Now, why exactly three? Two objections come up, and both deserve an answer. The first: could the curve run along the axis for a while?
That would need it to return zero at every input in a stretch, and no quadratic with a leading number does that. It was checked across all six hundred and forty-eight, and not one of them returns zero everywhere. The second objection is about the drawing rather than the curve. Could a curve that looks like it misses come back down and cross, if you extended the picture far enough?
No, and this is where the single turn earns its keep. Past the turn, an upward-opening curve only climbs. There is no coming back. Extending the drawing shows you more of the same arms going further away from the axis, and never a fourth thing. Which leaves the result the three cases were for. A quadratic meets the horizontal axis at most twice. At most, not exactly: the first case reaches two, the second case equals it with two that coincide, and the third case misses entirely.
Now, that sentence is easy to say dishonestly, so here is how it was actually checked. The number of meetings was counted three separate ways: by walking along the curve and watching which side of the axis it was on, by an exact method that works on a polynomial of any degree, and by the completed square. All three agreed on all six hundred and forty-eight. But agreeing is not the interesting part.
A counter that can only ever answer nought, one or two will report a ceiling of two no matter what you feed it, and that would prove nothing. So the first two counters were handed cubics, through exactly the same lines of code, and both came back with three. Then a curve of degree four, and both came back with four. Only then does their refusal to say three about a quadratic mean anything.
It never happened. The largest number either of them ever reported across the whole sweep was two. So, two questions and two answers. Which way does it open? Look at the sign of the number multiplying the square, and you are done. How many times does it meet the axis? Twice, once, or not at all, and there is no fourth answer. Which also tells you how many zeroes there are: two, two that coincide, or none.
Notice what has not been given. Nothing here tells you in advance which of the three cases a particular quadratic falls into. You can see it once the curve is drawn, and you can sometimes argue it by hand, as we did with x squared minus x plus one. But there is no test yet, and that is honest rather than unfinished. What you do have is a ceiling, and a reason for it.
One bend, two arms, one axis. Three ways they can meet, and no more.
Where this fits
Taken from the notes each video was made from, not from the reading order — these are the ideas this one rests on and the ones that later rest on it.
Builds on
- A zero is exactly a point where the curve meets the horizontal axisClass 10 · Ch 2, Polynomials
- Degree as the label that separates linear, quadratic and cubicClass 10 · Ch 2, Polynomials
Comes up again in
- Why degree puts a ceiling on how many zeroes there can beClass 10 · Ch 2, Polynomials