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# 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

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Mathematics
(Chapter – 2) (Polynomials)
(Class – X)

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:

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
(Chapter – 2) (Polynomials)
(Class – X)

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:

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
(Chapter – 2) (Polynomials)
(Class – X)

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
(Chapter – 2) (Polynomials)
(Class – X)

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:

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
(Chapter – 2) (Polynomials)
(Class – X)

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
(Chapter – 2) (Polynomials)
(Class – X)

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.

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.

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
(Chapter – 2) (Polynomials)
(Class – X)

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:

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
(Chapter – 2) (Polynomials)
(Class – X)

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
(Chapter – 2) (Polynomials)
(Class – X)

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.

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.

Zeroes are a - b, a + a + b
Comparing the given polynomial with , we obtain
p = 1, q = -3, r = 1, t = 1
are
Mathematics
(Chapter – 2) (Polynomials)
(Class – X)

4

Hence, a = 1 and b =

Question 4:
]It two zeroes of the polynomial ,
find other zeroes.

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
(Chapter – 2) (Polynomials)
(Class – X)

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:

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
(Chapter – 2) (Polynomials)
(Class – X)

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
(Chapter – 2) (Polynomials)
(Class – X)

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.

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.

Zeroes are a - b, a + a + b
Comparing the given polynomial with , we obtain
p = 1, q = -3, r = 1, t = 1
are
Mathematics
(Chapter – 2) (Polynomials)
(Class – X)

4

Hence, a = 1 and b =

Question 4:
]It two zeroes of the polynomial ,
find other zeroes.

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
(Chapter – 2) (Polynomials)
(Class – X)

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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