NAVODAYA VIDYALA SAMITI

NOIDA, UP

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E- CONTENT

MATHEMATICS

CLASS X

CHAPTER 2

POLYNOMIALS

Geometrical meaning of the zeroes of a polynomial:-

We have seen that a polynomial takes different values

for different values of the variable.

Consider the example p(x) = 2x-4.

We can have the different values as follows.

X : 1 2 3 0 -1 1/2

P(x) : -2 0 2 -4 -6 -3

By taking the values of the variable x and the corresponding values of the polynomial p(x) as ordered pairs, we can plot the points on a Cartesian co-ordinate system.

By joining such points corresponding to the given

polynomial, we get the graph of the given polynomial.

The different ordered pairs corresponding to the

polynomial 2x – 4 are (1,-2), (2,0), (3,2), (0, -4),

(-1, -6), (1/2 , -3) etc.

Let us see the graph of the above polynomial.

Note that the graph of the given first degree polynomial

2x – 4 is a straight line and hence the name

linear polynomial (representing a line).

The zero of this polynomial is x= 2, which is the

x- coordinate of the point where the line

meets the x- axis.

This is because of the fact that the value of the

polynomial is taken as the y co-ordinate of each point

and the y co-ordinate of a point on the X- axis is 0.

In general , the x co-ordinate of the point where

the graph of a polynomial meets the X- axis is the zero of the polynomial.

In the same way , we can plot the graph of a quadratic

polynomial.

For example, p(x)= x²- 4.

X : 1 -1 2 -2 3

P(x): -3 -3 0 0 5

The corresponding

points are (1,-3), (-1, -3),(2,0),

(-2, 0), (3,5) etc. By plotting

these points , we can get

the graph as

Here, we can see that there are two points where the graph

meets the x- axis and hence the zeroes of the polynomial are

x = 2 and -2.

Thus a quadratic polynomial can have at most two zeroes.

Note:- The shape of the graph of a quadratic polynomial

is a curve known as Parabola which can be opened upwards ( U)

or opened downwards ( ∩ ).

The number of zeroes will be the number of points where the

Graph meets the X- axis and the zeroes are the

x- co-ordinates of the point(s) where the graph meets

the X- axis.

PREPARED BY

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JOAN A LUKE

JNV KOLLAM

KERALA HYDERABAD REGION