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Polynomials

Graphical Interpretation of Number of Zeros of a Polynomial

We have learnt to find the zeroes or roots of a polynomial. Also, we know that zeroes or roots of a polynomial are those values of its variables for which the polynomial results to zero.      

Now, let us learn about the relation between zeroes and the coefficients of the polynomial.

Relationship between the zero and the coefficient of a linear polynomial:

Let p(x) = ax + b be a linear polynomial such that a ≠ 0. 

Zero of this polynomial can be obtained by equating it with 0.

i.e,  p(x) = 0

⇒ ax + b = 0

⇒ ax = –b 

Using this relation, we can find the zero of a linear polynomial.

 

Relationship between the zeroes and the coefficients of a quadratic polynomial:

Consider a quadratic polynomial p(x) = 3x2– 5x – 12.

Can we find out the sum and the product of the zeroes of this polynomial?

Yes, we can find the sum and the product of zeroes but firstly we have to find out the zeroes of the polynomial.

Here, the zeroes of polynomial p(x) are 3 and.

Now, the sum of zeroes = 3 +

And the product of zeroes = 3 × = – 4

Can we find out the sum and the product of zeroes by any other method?

Yes, there is also a method in which there is no need to find out the zeroes. In that method we use the coefficients of the polynomial to find the sum and the product of zeroes.

Firstly let us see the relation between the sum and product of zeroes and the coefficients of the polynomial.

Let us first consider a quadratic polynomial p(x) = ax2 + bx + c, where a, b and c are constants.

If αand β are the zeroes of p(x) = ax2 + bx + c, then,

Sum of zeroes = and

Product of zeroes = αβ =

Now, let us find the sum and product of zeroes of the polynomial given in the beginning, using these relations.

The polynomial is p(x) = 3x2 – 5x – 12.

On comparing this equation with ax2 + bx + c, we have

a = 3, b = –5 and c = –12

∴ Sum of zeroes = –=

And the product of zeroes =

Using these relations we obtained the same values as we found after calculating the zeroes.

Now, let us know the relations between the sum and the product of zeroes and the coefficients of a cubic polynomial.

The general form of a cubic polynomial is p(x) = ax3 + bx2 + cx + d where a, b, c and d are constants.

Ifα, β and γ are the three zeroes of cubic polynomial p(x), then the relations are given by

Let us solve some more problems to have a better understanding of the concept.

Example 1:

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

(a) x2 – 9x + 20 (b) 5x2 + 21x – 20.

Solution:

(a) The given quadratic polynomial is p(x) = x2 – 9x + 20.

Firstly we will find the zeroes by the method of splitting the middle term and then verify the relationship between the zeroes and the coefficients of the polynomial.

In this method, we have to find two numbers whose product is 20 and sum (or difference) is 9.

Such two numbers are 4 and 5 so, we have

x2 – 9x + 20 = x2 – (4 + 5)x + 20

= x2 – 4x – 5x + 20

= x (x – 4) – 5 (x – 4)

= (x – 4) (x – 5)

Thus, for finding zeroes, we put x2 – 9x + 20 = 0.

(x – 4) (x – 5) = 0

(x – 4) = 0 or (x – 5) = 0

x = 4 or 5

Thus, 4 and 5 are the zeroes of the polynomial p(x) = x2 – 9x + 20.

Now sum of zeroes = 4 + 5 = 9

And product of zeroes = 4 × 5 = 20

Now, on comparing p(x) = x2 – 9x + 20 with the general quadratic polynomial

ax2 + bx + c = 0, we have a = 1, b = –9 and c = 20.

Using the formulae, we have

Sum of zeroes

= 9

Product of zeroes

= 20

Hence, it is verified that Sum of zeroes = and

Product of zeroes = αβ = .

(b) Comparing the given polynomial with ax2 + bx + c, we have, a = 5, b = 21

and c = – 20.

Using the method of splitting the middle term we have,

p(x) = 5x2 + 21 x – 20 = 5x2 + (25 – 4)x – 20

= 5x2 + 25x –4 x – 20

= 5x(x + 5) –4 (x + 5)

= (x + 5) (5x – 4)

For finding zeroes, put p(x) = 0

(x + 5) (5x – 4) = 0

x = – 5 or x =

Thus, the zeroes of the polynomial 5x2 + 21 x – 20 are – 5 and.

Now, sum of zeroes = – 5 + = =

And product of zeroes = – 5 × = –4 = =

Hence, verified.

Example 2: 

If 1, 2, and 6 are the zeroes of the cubic polynomial x3 – 9x2 + 20x – 12, then verify the relations between the zeroes and the coefficients.

Solution:

The given cubic polynomial is p(x) = x3 – 9x2 + 20x – 12.

On comparing the given polynomial with the general form ax3 …

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