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Polynomials

Graphical Interpretation of Number of Zeros of a Polynomial

Let us consider a polynomial p(x) = 2x – 5. Its graph can be drawn as follows:

If we notice the above graph carefully, then we can see that the graph of the polynomial p(x) = 2x – 5 passes through the x-axis at a point.

Do you know what is the value of the polynomial p(x) at the x-coordinate of this point?

Let us find out how.

Thus, is a zero of the polynomial.

Thus, we can say that

The x-coordinates of the points at which the graph of the polynomial intersects the x-axis are the zeroes of the polynomial.

Using the above statement, we can find out the zeroes of the polynomials. The method of finding zeroes can be easily understood with the help of the following example.

Let us find the zeroes of the polynomial p(x) = x2 – 4x + 3. The graph of this polynomial can be drawn as follows:

From the graph, we can see that the graph of the polynomial p(x) = x2 – 4x + 3 cuts the x-axis at two points respectively (1, 0) and (3, 0). Now, the x-coordinates of these points are 1 and 3 respectively. Thus, 1 and 3 are the zeroes of the given polynomial p(x) = x2 – 4x + 3.

In this way we can interpret the zeroes of the polynomial geometrically.

Let us discuss one more example based on geometrical interpretation of zeroes of a polynomial.

Example1:

Find the zeroes of the polynomials whose graphs are shown below.

Solution:

(i) The graph of the polynomial intersects the x-axis at the point (2, 0).

The x coordinate of the point (2, 0) is 2.

Thus, 2 is a zero of the polynomial whose graph has been shown.

(ii) The graph of the polynomial intersects the x-axis at the point (0, 0).

The x co-ordinate of the point (0, 0) is 0.

Thus, 0 is a zero of the polynomial.

(iii) The graph does not intersect the x-axis, so there is no zero of the polynomial whose graph has been shown.

5 and –3 are the zeroes of a quadratic polynomial p(x). Can we form the quadratic polynomial with the help of these zeroes?

Yes, we can. Let us see how we can form the polynomial.

5 is a zero of the polynomial, it means that (x – 5) is a factor of the polynomial p(x). Similarly (x + 3) is also a factor of the polynomial p(x). We know that a quadratic polynomial can have only two linear factors. Thus, we can write the polynomial p(x) as follows:

p(x) = (x – 5) (x + 3)

= x2 + (– 5 + 3) x + (– 5) (3)

[Using the identity (x + a) (x + b) = x2 + (a + b) x + ab, where a = – 5 and b = 3]

Thus, p(x) = x2 – 2x – 15

This is the required quadratic polynomial.

If we know the zeroes of the quadratic polynomial, then we can find the polynomial as above. On the other hand, if we know the sum and product of the zeroes of the polynomial, then we can use the following formula to find the polynomial.

p(x) = x2 – (sum of zeroes)x + product of zeroes

For example: if the sum and the product of the zeroes of a polynomial are 2 and –8 respectively, then find the polynomial.

Using the above formula, we have

p(x) = x2 – (2)x + (–8)

p(x) = x2 – 2x – 8

In this way, we can find the polynomial if the relationship between the zeroes is known to us.

Let us discuss some more examples based on formation of quadratic polynomial using sum and products of zeroes.

Example 1:

If the sum and product of zeroes of a quadratic polynomial are 13 and 36 respectively, find the quadratic polynomial. Solution:

Let the polynomial be p(x).

It is given that the sum of the zeroes of the polynomial = 13 and the product of the zeroes of the polynomial = 36

But we know that the quadratic polynomial p(x) in terms of sum and product of zeroes is given by the formula

p(x) = x2 – (sum of zeroes)x + product of zeroes

so, we have

p(x) = x2 – 13x + 36

Therefore, x2 – 13x + 36 is the quadratic polynomial whose sum and the product of zeroes are 13 and 36 respectively.

Example 2:

If the sum and product of zeroes of a quadratic polynomial are and – 6 respectively, find the quadratic polynomial.

Solution:

Let the quadratic polynomial be p(x).

It is given that the sum of the zeroes of the polynomial is and the product of the zeroes of the polynomial is – 6.

We know that the quadratic polynomial p(x) in terms of sum and product of zeroes is given by the formula

p(x) = x2 – (sum of zeroes)x + product of zeroes

so, we have

p(x) = x2 – ()x + (– 6)

p(x) = x2 x – 6

Therefore, x2 x – 6 is the quadratic polynomial whose sum and product of zeroes are and – 6 respectively.

Example 3:

If and are the zeroes of a quadratic polynomial, then find…

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