Q. 14.2( 62 Votes )

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)

Answer :

(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 CoefficientsChamp Quiz |Revealing the relation Between Zero and CoefficientsChamp Quiz |Revealing the relation Between Zero and Coefficients38 mins
Relation Between zeroes and CoefficientsRelation Between zeroes and CoefficientsRelation Between zeroes and Coefficients46 mins
Interactive Quiz - Geometrical Meaning of the ZeroesInteractive Quiz - Geometrical Meaning of the ZeroesInteractive Quiz - Geometrical Meaning of the Zeroes32 mins
Relationship between Zeroes and Coefficients-2Relationship between Zeroes and Coefficients-2Relationship between Zeroes and Coefficients-238 mins
Quiz - Division AlgorithmQuiz - Division AlgorithmQuiz - Division Algorithm38 mins
Relationship between Zeroes and Coefficients-1Relationship between Zeroes and Coefficients-1Relationship between Zeroes and Coefficients-152 mins
Interactive Quiz:PolynomialsInteractive Quiz:PolynomialsInteractive Quiz:Polynomials43 mins
Division Algorithm-1Division Algorithm-1Division Algorithm-130 mins
Relation b/w The Zeroes and Coefficients of Cubic PolynomialsRelation b/w The Zeroes and Coefficients of Cubic PolynomialsRelation b/w The Zeroes and Coefficients of Cubic Polynomials54 mins
Solving Imp. Qs. of Olympiad on PolynomialsSolving Imp. Qs. of Olympiad on PolynomialsSolving 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
caricature
view all courses