## Relation Between The Zeros And The Coefficients Of A Polynomial

### 1. Quadratic polynomial: ; a ¹ 0.

Let a, b are two zeros of the given polynomial.

Sum of zeros (a + b) =

Product of zeros (ab) =

### 2. Cubic polynomial: ; a ¹ 0

Let a, b and g are three zeros of the given polynomial.

(i) Sum of zeros

(ii) Product of zeros taken two at a time

(ab + bg + ga) =

(iii) Product of zeros (abg) =

### 3. Formation of Quadratic Polynomial:

Let a, b are the zeros, then required polynomial is

k[x2 – (sum of roots)x + (product of roots)] or k[(x–a)(x–b)]

where k is a non-zero constant

### 4. Formation of Cubic Polynomial

Let a, b, g are the zeros then required polynomial is

or

where k is a non-zero constant

### Question:

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

(i) (ii) (iii)

### Solution:

**(i) Given quadratic polynomial is **

=

=

y = 0 gives x = –2, 4. These are two zeros such that a = –2, b = 4.

\ Sum of zeros (a + b) = –2 + 4 = 2 =

Product of zeros (ab) = – 2 × 4 = – 8 = =

**(ii) Given quadratic polynomial is **

y =

y = 0 gives. These are two zeros such that

Sum of zeros (a + b) =

Product of zeros (ab) = =

**(iii) **

y = 0 gives and ; i.e. u = 0, –2.

a =0, b = –2

Sum of zeros (a + b) = 0 – 2 = –2 = =

Product of zeros (ab) = 0 × (–2) = .

### Question:

Find a quadratic polynomial, the sum and product of whose zeros are and –1 respectively.

### Solution:

Here , ab = –1

\ Required polynomial is

= where k is a non-zero constant.

### Question:

Form a cubic polynomial with zeros a = 3, b = 2, g = –1 .

### Solution:

a = 3, b = 2, g = –1

Required polynomial =

=

=

=

=

where k is a non-zero constant.

### Question:

Find a quadratic polynomial whose zeros are 2 and –3.

### Solution:

Required polynomial

where k is a non-zero constant

### Question:

If a and b are the zeros of the polynomial such that

a – b = 1, find the value of k.

### Solution:

Since a and b are the zeros of the polynomial

\ and

Now, [Given]

Þ

Þ

Þ 25 – 4k = 1

Þ 24 = 4k

Þ k = 6

Hence, the value of k is 6.

### Question:

Verify that the numbers given alongside of the cubic polynomials below are the zeros. Also verify the relationship between the zeros and co-efficients in each case.

; 2, 1, 1

### Solution:

Let

On comparing with

, , c = 5, d = –2

Given zeros are 2, 1, 1.

\ 2, 1, 1 are zeros of

a = 2, b = 1, g = 1

=

= 2 + 1 +2 = =

Hence the result.

### Question:

Write a rational expression whose numerator is a quadratic polynomial with zeros 2 and –1 and denominator is a quadratic polynomial with zeros and 3.

### Solution:

Zeros of numerator are 2 and –1.

a = 2, b = –1, ,

Numerator is

Zeros of denominator are , 3.

, , ,

Denominator is

= where k = .

\ Rational expression is =

### Question:

Find a quadratic polynomial whose zeros are reciprocals of the zeros of the polynomial

### Solution:

Let a, b be the zeros of the polynomial Then,

and

Let S and P denote respectively the sum and product of the zeros of a polynomial whose zeros are and . Then,

and

Hence, the required polynomial is given by

where k is any non-zero constant.