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.