NCERT Solutions, Exercise 2.4 : Polynomials, Class 10 (Mathematics) Class 10 Notes | EduRev

Class 10 : NCERT Solutions, Exercise 2.4 : Polynomials, Class 10 (Mathematics) Class 10 Notes | EduRev

 Page 1


Mathematics 
(www.tiwariacademy.com) 
(Chapter – 2) (Polynomials)  
(Class – X) 
 
www.tiwariacademy.com  
1 
Exercise 2.4 
Question 1:  
Verify that the numbers given alongside of the cubic polynomials below 
are their zeroes. Also verify the relationship between the zeroes and the 
coefficients in each case:  
  
Answer 1:  
  
 
Therefore, ½ , 1, and -2 are the zeroes of the given polynomial.  
Comparing the given polynomial with ,  
we obtain a = 2, b = 1, c = -5, d = 2  
 (i)    
  
Page 2


Mathematics 
(www.tiwariacademy.com) 
(Chapter – 2) (Polynomials)  
(Class – X) 
 
www.tiwariacademy.com  
1 
Exercise 2.4 
Question 1:  
Verify that the numbers given alongside of the cubic polynomials below 
are their zeroes. Also verify the relationship between the zeroes and the 
coefficients in each case:  
  
Answer 1:  
  
 
Therefore, ½ , 1, and -2 are the zeroes of the given polynomial.  
Comparing the given polynomial with ,  
we obtain a = 2, b = 1, c = -5, d = 2  
 (i)    
  
Mathematics 
(www.tiwariacademy.com) 
(Chapter – 2) (Polynomials)  
(Class – X) 
 
www.tiwariacademy.com  
2 
  
Therefore, the relationship between the zeroes and the coefficients is 
verified.  
 
(ii)   
  
Therefore, 2, 1, 1 are the zeroes of the given polynomial.  
Comparing the given polynomial with ,  
we obtain a = 1, b = -4, c = 5, d = -2.  
Verification of the relationship between zeroes and coefficient of the 
given polynomial  
  
Multiplication of zeroes taking two at a time 
 
 = (2)(1) + (1)(1) + (2)(1) =2 + 1 + 2 = 5   
 
Multiplication of zeroes = 2 × 1 × 1 = 2   
  
Page 3


Mathematics 
(www.tiwariacademy.com) 
(Chapter – 2) (Polynomials)  
(Class – X) 
 
www.tiwariacademy.com  
1 
Exercise 2.4 
Question 1:  
Verify that the numbers given alongside of the cubic polynomials below 
are their zeroes. Also verify the relationship between the zeroes and the 
coefficients in each case:  
  
Answer 1:  
  
 
Therefore, ½ , 1, and -2 are the zeroes of the given polynomial.  
Comparing the given polynomial with ,  
we obtain a = 2, b = 1, c = -5, d = 2  
 (i)    
  
Mathematics 
(www.tiwariacademy.com) 
(Chapter – 2) (Polynomials)  
(Class – X) 
 
www.tiwariacademy.com  
2 
  
Therefore, the relationship between the zeroes and the coefficients is 
verified.  
 
(ii)   
  
Therefore, 2, 1, 1 are the zeroes of the given polynomial.  
Comparing the given polynomial with ,  
we obtain a = 1, b = -4, c = 5, d = -2.  
Verification of the relationship between zeroes and coefficient of the 
given polynomial  
  
Multiplication of zeroes taking two at a time 
 
 = (2)(1) + (1)(1) + (2)(1) =2 + 1 + 2 = 5   
 
Multiplication of zeroes = 2 × 1 × 1 = 2   
  
Mathematics 
(www.tiwariacademy.com) 
(Chapter – 2) (Polynomials)  
(Class – X) 
 
www.tiwariacademy.com  
3 
Hence, the relationship between the zeroes and the coefficients is 
verified.  
 
Question 2:  
Find a cubic polynomial with the sum, sum of the product of its zeroes 
taken two at a time, and the product of its zeroes as 2, - 7, - 14 
respectively.  
 
Answer 2:  
Let the polynomial be and the zeroes be 
 .  
It is given that  
  
If a = 1, then b = -2, c = -7, d = 14  
Hence, the polynomial is .  
 
Question 3:  
If the zeroes of polynomial , find a and b.  
 
Answer 3:  
  
Zeroes are a - b, a + a + b  
Comparing the given polynomial with , we obtain  
p = 1, q = -3, r = 1, t = 1  
are 
Page 4


Mathematics 
(www.tiwariacademy.com) 
(Chapter – 2) (Polynomials)  
(Class – X) 
 
www.tiwariacademy.com  
1 
Exercise 2.4 
Question 1:  
Verify that the numbers given alongside of the cubic polynomials below 
are their zeroes. Also verify the relationship between the zeroes and the 
coefficients in each case:  
  
Answer 1:  
  
 
Therefore, ½ , 1, and -2 are the zeroes of the given polynomial.  
Comparing the given polynomial with ,  
we obtain a = 2, b = 1, c = -5, d = 2  
 (i)    
  
Mathematics 
(www.tiwariacademy.com) 
(Chapter – 2) (Polynomials)  
(Class – X) 
 
www.tiwariacademy.com  
2 
  
Therefore, the relationship between the zeroes and the coefficients is 
verified.  
 
(ii)   
  
Therefore, 2, 1, 1 are the zeroes of the given polynomial.  
Comparing the given polynomial with ,  
we obtain a = 1, b = -4, c = 5, d = -2.  
Verification of the relationship between zeroes and coefficient of the 
given polynomial  
  
Multiplication of zeroes taking two at a time 
 
 = (2)(1) + (1)(1) + (2)(1) =2 + 1 + 2 = 5   
 
Multiplication of zeroes = 2 × 1 × 1 = 2   
  
Mathematics 
(www.tiwariacademy.com) 
(Chapter – 2) (Polynomials)  
(Class – X) 
 
www.tiwariacademy.com  
3 
Hence, the relationship between the zeroes and the coefficients is 
verified.  
 
Question 2:  
Find a cubic polynomial with the sum, sum of the product of its zeroes 
taken two at a time, and the product of its zeroes as 2, - 7, - 14 
respectively.  
 
Answer 2:  
Let the polynomial be and the zeroes be 
 .  
It is given that  
  
If a = 1, then b = -2, c = -7, d = 14  
Hence, the polynomial is .  
 
Question 3:  
If the zeroes of polynomial , find a and b.  
 
Answer 3:  
  
Zeroes are a - b, a + a + b  
Comparing the given polynomial with , we obtain  
p = 1, q = -3, r = 1, t = 1  
are 
Mathematics 
(www.tiwariacademy.com) 
(Chapter – 2) (Polynomials)  
(Class – X) 
 
www.tiwariacademy.com  
4 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
Hence, a = 1 and b =  
 
Question 4:  
]It two zeroes of the polynomial , 
find other zeroes.  
 
Answer 4:  
Given 2 + v3 and 2 – v3  are zeroes of the given polynomial.  
So, (2 + v3)(2 – v3)   is a factor of polynomial. 
Therefore,[x – (2 + v3)][x – (2 – v3)]  = x
2
 + 4 - 4x - 3  
= x
2
 - 4x + 1 is a factor of the given polynomial  
For finding the remaining zeroes of the given polynomial, we will find  
  
The zeroes are  .  
or  
are 
Page 5


Mathematics 
(www.tiwariacademy.com) 
(Chapter – 2) (Polynomials)  
(Class – X) 
 
www.tiwariacademy.com  
1 
Exercise 2.4 
Question 1:  
Verify that the numbers given alongside of the cubic polynomials below 
are their zeroes. Also verify the relationship between the zeroes and the 
coefficients in each case:  
  
Answer 1:  
  
 
Therefore, ½ , 1, and -2 are the zeroes of the given polynomial.  
Comparing the given polynomial with ,  
we obtain a = 2, b = 1, c = -5, d = 2  
 (i)    
  
Mathematics 
(www.tiwariacademy.com) 
(Chapter – 2) (Polynomials)  
(Class – X) 
 
www.tiwariacademy.com  
2 
  
Therefore, the relationship between the zeroes and the coefficients is 
verified.  
 
(ii)   
  
Therefore, 2, 1, 1 are the zeroes of the given polynomial.  
Comparing the given polynomial with ,  
we obtain a = 1, b = -4, c = 5, d = -2.  
Verification of the relationship between zeroes and coefficient of the 
given polynomial  
  
Multiplication of zeroes taking two at a time 
 
 = (2)(1) + (1)(1) + (2)(1) =2 + 1 + 2 = 5   
 
Multiplication of zeroes = 2 × 1 × 1 = 2   
  
Mathematics 
(www.tiwariacademy.com) 
(Chapter – 2) (Polynomials)  
(Class – X) 
 
www.tiwariacademy.com  
3 
Hence, the relationship between the zeroes and the coefficients is 
verified.  
 
Question 2:  
Find a cubic polynomial with the sum, sum of the product of its zeroes 
taken two at a time, and the product of its zeroes as 2, - 7, - 14 
respectively.  
 
Answer 2:  
Let the polynomial be and the zeroes be 
 .  
It is given that  
  
If a = 1, then b = -2, c = -7, d = 14  
Hence, the polynomial is .  
 
Question 3:  
If the zeroes of polynomial , find a and b.  
 
Answer 3:  
  
Zeroes are a - b, a + a + b  
Comparing the given polynomial with , we obtain  
p = 1, q = -3, r = 1, t = 1  
are 
Mathematics 
(www.tiwariacademy.com) 
(Chapter – 2) (Polynomials)  
(Class – X) 
 
www.tiwariacademy.com  
4 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
Hence, a = 1 and b =  
 
Question 4:  
]It two zeroes of the polynomial , 
find other zeroes.  
 
Answer 4:  
Given 2 + v3 and 2 – v3  are zeroes of the given polynomial.  
So, (2 + v3)(2 – v3)   is a factor of polynomial. 
Therefore,[x – (2 + v3)][x – (2 – v3)]  = x
2
 + 4 - 4x - 3  
= x
2
 - 4x + 1 is a factor of the given polynomial  
For finding the remaining zeroes of the given polynomial, we will find  
  
The zeroes are  .  
or  
are 
Mathematics 
(www.tiwariacademy.com) 
(Chapter – 2) (Polynomials)  
(Class – X) 
 
www.tiwariacademy.com  
5 
  
  
 
It can be observed that polynomial  
Therefore, the value of the polynomial is also zero when or  
  
Or x = 7 or -5  
Hence, 7 and -5 are also zeroes of this polynomial.  
  
 
 Question 5:  
If the polynomial is divided by another 
polynomial, the remainder comes out to be x + a, find k and 
a.  
 
  
Clearly,  =   
is also a factor of the given  
= 
  
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