Geometrical Meaning of the Zeroes of a Polynomial

Geometrical meaning of a zero of a linear polynomial

Why are the zeroes of a polynomial so important?

To understand that, first lets look at the geometrical representation of linear and quadratic polynomials and the geomterical meaning of their zeroes.

When you just look at the equations with x and y, polynomials may appear daunting for some. But polynomials become beautiful when you represent them geometrically. Once you are able to the polynomial equation on a graph you can understand the behaviour of the problem in a much better way.

For example, lets look at the equation y = 2x + 3. What does this mean geometrically? This looks very .

But if we substitute different values for x and get y we get multiple points in the co-ordinate system.

Let us try doing the substition of values of in the equation y = 2x + 3 and find the value of .

x	-2	2
y = 2x+3	-1	7

When we plot these points in the co-ordinate system we get a graph like below. This makes it much easier to visualize the equation and we can imagine how the different values of x are going to affect y.

Is it better to view polynomials in graph form?;Yes/No

From the graph we can see that the graph of y = 2 x + 3 intersects the x - axis at

You also know that the zero of 2x + 3 is −32. Thus, the zero of the polynomial 2 x + 3 is the x-coordinate of the point where the graph of y = 2x + 3 intersects the x-axis.

The graph of y=2x+3 intersects the x-axis in place(s).

The linear polynomial ax + b, a 0, has exactly one zero, namely, the x-coordinate of the point where the graph of y = ax + b intersects the x-axis, which is the point (−ba,0).

Geometrical meaning of a zero of a quadratic polynomial

Consider the quadratic polynomial x2−3x−4. Let us see what the graph of y=x2−3x−4 looks like. To do that, lets list out the values of y for different values of x.

x	-2	-1	0	1	2	3	4	5
y=x2−3x−4

Now let's plot the graph with these values.

Graph ploting of the quadratic polynomial x2−3x−4 is

From the table above, the zeros of the quadratic polynomial y=x2−3x−4 are and .

In fact, for any quadratic polynomial ax2+bx+c, a ≠ 0, the graph of the corresponding equation y = ax2+bx+c has one of the two shapes either open upwards like ◡ or open downwards like ◠ depending on whether a > 0 or a < 0. (These curves are called parabolas.)

For any quadratic polynomial, i.e., the zeroes of a quadratic polynomial ax2+bx+c, a ≠ 0, are precisely the of the points where the parabola representing y = ax2+bx+c intersects the x-axis.

From the above examples we observe that there are 3 cases, any quadratic equation will have 0 or 1 or 2 zeroes.

Case(i)The graph cuts x-axis at two distinct points where the x-coordinates are the two zeroes of the quadratic polynomial ax2+ bx + c in this case.

Case(ii)The graph intersects the x-axis at only one point, or at two coincident points. The x-coordinates is the one zero for the quadratic polynomial ax2+ bx + c in this case.

Case(iii)The x-coordinates is the no zero for the quadratic polynomial ax2+ bx + c in this case.

Graphs are given in the below step. Drop each of them into concerned boxes.

Instructions

Case (i)

Case (ii)

Case (iii)

So, you can see geometrically that a quadratic polynomial can have either two distinct zeroes or two equal zeroes (i.e., one zero), or no zero. This also means that a polynomial of degree 2 has atmost zeroes.

Geometrical meaning of a zero of a cubic polynomial

Lets look at some cubic polynomials and plot them to get their geometrical meaning. From that let's try to find some properties.Consider the cubic polynomial x3−4x. To see what the graph of y = x3−4x looks like, let us list a few values of y corresponding to a few values for x as shown in the table below:

Instructions

x	-2	-1	0	1	2
y=x3−4x

(i) y=x3−4x

We see from the table above that – 2, 0 and 2 are zeroes of the cubic polynomial x3−4x. Observe that – 2, 0 and 2 are, in fact, the x-coordinates of the only points where the graph of y = x3−4xintersects the .

Since the curve meets the x-axis in only these 3 points, their x-coordinates are the only zeroes of the polynomial. Let us take a few more examples.

(ii)y=x3

Move your mouse over the graph and find the zero polynomial of this graph. This has zero polynomials.

(iii) y=x3−x2

Move your mouse over the graph and find the zero polynomial of this graph. This has zero polynomials.

Note that is the only zero of the polynomial x3 .

We can see that 0 is the x-coordinate of the only point where the graph of y = x3 intersects the x-axis.

Similarly, since x3−x2 = x2x−1, and are the only zeroes of the polynomial x3−x2. These values are the x - coordinates of the only points where the graph of y=x3−x2 intersects the x-axis.

From the examples above, we see that there are at most 3 zeroes for any cubic polynomial. In other words, any polynomial of degree 3 can have at most three zeroes.

Remark : In general, given a polynomial p(x) of degree n, the graph of y = p(x) intersects the x-axis at atmost n points. Therefore, a polynomial p(x) of degree n has at most zeroes.

Look at the graphs in figure given below. Each is the graph of y = p(x), where p(x) is a polynomial. For each of the graphs, find the number of zeroes of p(x).

Solution :