# Find the zeros of each of the following quadratic polynomials and verify the relationship between the zeros and their coefficients:(i) (ii) (iii) (iv) (v) (vi)(vii) (viii)

(i)
factorize the given polynomial by splitting the middle term:

⇒ x2 - 4x + 2x – 8

⇒ x (x - 4) + 2 (x - 4)

For zeros of f(x),
f(x) = 0

⇒(x + 2) (x - 4) = 0
x+2=0
x=-2
x-4=0
x=4

⇒x = -2, 4

Therefore zeros of the polynomial are -2 & 4
In a polynomial the relations hold are as follows:
sum of zeroes is equal to
product of zeroes is equal to
For the given polynomial,

Sum of zeros = -2 + 4 = 2
And   is -(-2) = 2
Hence the value of  and sum of zeroes are same.

Product of zeros = -2 × 4 = -8
is -8.
Hence the value of  and product of zeroes are same.

(ii)
factorize the given polynomial by splitting the middle term:

⇒ 4s2 -2s - 2s + 1

⇒ 2s (2s - 1) -1 (2s - 1)

For zeros of g(s), g(s) = 0

(2s - 1) (2s - 1) = 0
2s - 1=0

s =

Therefore zeros of the polynomial are ,
In a polynomial the relations hold are as follows:
sum of zeroes is equal to
product of zeroes is equal to
For the given polynomial,

Sum of zeros = + = 1

Hence the value of  and sum of zeroes are same.
Product of zeros = × =

Hence the value of  and product of zeroes are same.

(iii)
use the formula to solve the above equation,
Here a is t and b is .
Solve the given expression as:

For zeros of h(t),
h(t) = 0

Therefore zeros of the given polynomial are t = √15 & -√15
In a polynomial the relations hold are as follows:
sum of zeroes is equal to
product of zeroes is equal to
For the given polynomial,

Sum of zeros = √15 + (- √15) = 0
The value of  is 0.
Hence, the value of  and sum of zeroes are same.

Product of zeros
The value of  is -.
Hence the value of  and product of zeroes are same.

(iv) f(x) =
Write the equation in the form of ax2 +bx+c as:

6x2 - 7x -3
factorize the given polynomial by splitting the middle term:

⇒ 6x2 - 9x + 2x - 3

⇒ 3x(2x - 3) +1(2x - 3)

⇒ (3x + 1) (2x - 3)

For zeros of f(x),
f(x) = 0

⇒ (3x + 1) (2x - 3) = 0

x =

Therefore zeros of the polynomial are
In a polynomial the relations hold are as follows:
sum of zeroes is equal to
product of zeroes is equal to

Sum of zeros = = = =

Product of zeros = × = = =

(v)

P (x) = x2 + 3√2x - √2x - 6

For zeros of p(x), p(x) = 0

⇒ x (x + 3√2) -√2 (x + 3√2) = 0

⇒ (x - √2) (x + 3√2) = 0

x = √2, -3√2

Therefore zeros of the polynomial are √2 & -3√2

Sum of zeros = √2 -3√2 = -2√2 = -2√2 =

Product of zeros = √2 × -3√2 = -6 = -6 =

(vi) q (x) = √3x2 + 10x + 7√3

⇒ √3x2 + 10x + 7√3

⇒ √3x2 + 7x + 3x + 7√3

⇒ √3x (x +) + 3 (x + )

⇒ (√3x + 3) (x +)

For zeros of Q(x), Q(x) = 0

(√3x + 3) (x +) = 0

X = ,

Therefore zeros of the polynomial are ,

Sum of zeros = +

Product of zeros = = × = 7 =

(vii) f(x) = x2 - (√3 + 1)x + √3

f(x) = x2 - √3x - x + √3

f(x) = x(x - √3) -1(x - √3)

f(x) = (x - 1) (x - √3)

For zeros of f(x), f(x) = 0

(x - 1) (x - √3) = 0

X = 1, √3

Therefore zeros of the polynomial are 1 & √3

Sum of zeros = 1 + √3 = √3 + 1=

Product of zeros = 1 × √3 = √3=

(viii) g(x) = a(x2 + 13) – x(a2 + 1)

g(x) = ax2 - a2x – x + a
g(x) = ax2 - (a2 + 1)x + a

g(x) = ax(x - a) -1(x - a)

g(x) = (ax - 1) (x - a)

For zeros of g(x), g(x) = 0

(ax - 1) (x - a) = 0

X = , a

Therefore zeros of the polynomial are & a

Sum of zeros

Product of zeros = × a = 1 = 1 =

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Champ Quiz |Revealing the relation Between Zero and Coefficients38 mins
Relation Between zeroes and Coefficients46 mins
Interactive Quiz - Geometrical Meaning of the Zeroes32 mins
Relationship between Zeroes and Coefficients-238 mins
Quiz - Division Algorithm38 mins
Relationship between Zeroes and Coefficients-152 mins
Interactive Quiz:Polynomials43 mins
Division Algorithm-130 mins
Relation b/w The Zeroes and Coefficients of Cubic Polynomials54 mins
Solving Imp. Qs. of Olympiad on Polynomials47 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
view all courses