PolynomialsGeometrical Meaning of the Zeroes of a Polynomial

Think back to the dotted trail of Aman's ball. It started on the ground at , curved up, and came back down to the ground at — exactly at the zeroes of . That is no coincidence.

Every polynomial draws a picture: the graph of . If is a zero, then , so the point is on the graph — and every point whose -coordinate is 0 lies on the . Let's test this idea on lines, then on parabolas, then on cubics.

Lines Cross Once

Start with a linear polynomial , where . You know from Class IX that the graph of is a . For example, the graph of passes through these two points:

x	−2	2
y=2x+3	−1	7

The blue line meets the -axis halfway between −2 and −1, at the red dot . And the zero of is — the very same number!

Change and in to hit all three red flags.

${a} + ${b}

Fig. 2.1 — the line , starting as . The red dot marks its zero; the flags are your targets.

Every flag you hit was at . In general, the line (with ) crosses the -axis at exactly one point, (). So a linear polynomial has exactly one zero: the -coordinate of the point where its line crosses the -axis.

Plot a Parabola Yourself

Now a quadratic: . To see its graph, work out for a few values of , as in Table 2.1:

Table 2.1 — values of

x	−2	−1	0	1	2	3	4	5
y	6	0	−4	−6	−6	−4	0	6

Exam note: you will not be asked to draw graphs of quadratic or cubic polynomials, and they are not marked. They are here to help you understand zeroes.

Tap the grid to plot all eight points of Table 2.1, then see what shape they make.

Fig. 2.2 — the graph of .

Joined up smoothly, the eight points give this curve — Fig. 2.2.

From Table 2.1, when is — the same zeroes the machine found. On Fig. 2.2 they are exactly the two points where the curve meets the -axis.

This curve is called a parabola. The graph of every quadratic (with ) is a parabola: it opens upwards like ∪ when , and downwards like ∩ when . And for every quadratic, the zeroes are the -coordinates of the points where the meets the -axis.

Why up or down?

When is very large — positive or negative — the term is far bigger than . If that term is hugely positive, so both arms of the curve shoot up. If both arms plunge down. The sign of alone decides which way the parabola opens.

Two, One or No Zeroes

A parabola is always a ∪ or a ∩, so it can meet the -axis in only three ways (Figs. 2.3–2.5). Open each tab — each one shows a ∪ and a ∩ version of the same case:

Case (i)

Both curves — one opening up, one opening down — cut the -axis at two different points, A and A′. The -coordinates of A and A′ are the two zeroes of .

Case (ii)

Here the two points have slid together into a single point A, where each curve just touches the axis. The curve meets the -axis at only one point — think of it as two points that coincide.

Case (iii)

These curves stay completely above, or completely below, the -axis. They never meet it, so has no zero in this case.

In Case (ii), the of the touching point A is the only zero of the polynomial.

Now park this parabola in each of the three cases.

− ${k}

So a quadratic polynomial has either two distinct zeroes, two equal zeroes (that is, one zero), or no zero at all. In other words, a polynomial of degree has at most two zeroes.

Cubics Can Wiggle

What do zeroes look like for a cubic polynomial? Take , and list a few values of as in Table 2.2:

Table 2.2

x	−2	−1	0	1	2
y=x3−4x	0	3	0	−3	0

Plot these points, join them smoothly, and you get the curve in Fig. 2.6: it rises, dips and rises again — a wiggle no parabola can make.

Fig. 2.6 — the graph of , with its zeroes marked.

From the table, the zeroes of are . On Fig. 2.6 these are exactly the -coordinates of the three points where the curve crosses the -axis — and it crosses nowhere else, so these are its only zeroes.

Not every cubic crosses three times. Here are the graphs of y = x³ (Fig. 2.7) and y = x³ − x² (Fig. 2.8):

Fig. 2.7 — y = x³

Fig. 2.8 — y = x³ − x²

The only zero of is 0, and the first curve crosses the -axis at just one point: the origin. For the second one, , so its zeroes are . The second curve just touches the axis at 0 and crosses it at 1.

Now slide this cubic roller-coaster up and down.

+ ${k}

Try as you like — you can never make a cubic meet the -axis more than three times. A polynomial of degree has at most three zeroes.

The pattern continues: a line (degree 1) meets the -axis at most once, a parabola (degree 2) at most twice, and a cubic (degree 3) at most three times.

Remark: In general, if is a polynomial of degree , the graph of meets the -axis at points. So a polynomial of degree has at most zeroes.

Reading Zeroes off a Graph

Example 1: Each graph in Fig. 2.9 is the graph of for some polynomial . How many zeroes does have in each case?

Count the points where each graph meets the -axis — a point where the curve only touches the axis counts too — and fill in the number under the graph.

(i) zeroes:

(ii) zeroes:

(iii) zeroes:

(iv) zeroes:

(v) zeroes:

(vi) zeroes:

Fig. 2.9

Solution: (i) one crossing, so one zero; (ii) two; (iii) three; (iv) a straight line crosses just once; (v) the curve only touches the axis at one point — and a touching point still counts as a zero; (vi) four crossings, four zeroes.

Speed Round: Count the Zeroes

Now train your eye on random graphs. Count the points where each curve meets the -axis — and remember, touching counts!

Get five right to move on.

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