CBSE Tutorials

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.