CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials

CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials to check each and every concept and their solutions are also provided in this website. In this solutions are useful for the Central Board Of Secondary Education Students to prepare our examinations well (सीबीएसई कक्षा 10 एनसीईआरटी समाधान). So, we can check once in this article and go to our preparation. You can download the CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials PDF is also available in this article.

Download the CBSE Class 10 NCERT Solutions for Maths chapter 2, Polynomials

Download the Class 10 Maths NCERT solutions for chapter 2, Polynomials- Click Here

CBSE Class 10 Maths NCERT Problems & Solutions

Question 1:

The graphs of y = p(x) are given in the following the figure, for some polynomials p(x). Find the number of zeroes of p(x), in each case.

(i)

CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 1

(ii)

(iii)

CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 2

(iv)

(v)

CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 3

(v)

ANSWER:

(i) The number of zeroes is 0 as the graph does not cut the x-axis at any point.

(ii) The number of zeroes is 1 as the graph intersects the x-axis at only 1 point.

(iii) The number of zeroes is 3 as the graph intersects the x-axis at 3 points.

(iv) The number of zeroes is 2 as the graph intersects the x-axis at 2 points.

(v) The number of zeroes is 4 as the graph intersects the x-axis at 4 points.

(vi) The number of zeroes is 3 as the graph intersects the x-axis at 3 points.

PAGE NO 33:

Question 1:

Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 4 CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 5

CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 6 

ANSWER:

CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 7

The value of is zero when x − 4 = 0 or + 2 = 0, i.e., when x = 4 or x = −2

Therefore, the zeroes of CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 8are 4 and −2.

Sum of zeroes = 

Product of zeroes CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 9

The value of 4s2 − 4s + 1 is zero when 2s − 1 = 0, i.e.,CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 10

Therefore, the zeroes of 4s2 − 4s + 1 areandCBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 11.

Sum of zeroes = 

Product of zeroes CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 12

The value of 6x2 − 3 − 7x is zero when 3x + 1 = 0 or 2− 3 = 0, i.e., CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 13or

Therefore, the zeroes of 6x2 − 3 − 7x areCBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 14.

Sum of zeroes = 

Product of zeroes = CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 15

The value of 4u2 + 8u is zero when 4u = 0 or u + 2 = 0, i.e., u = 0 or u = −2

Therefore, the zeroes of 4u2 + 8u are 0 and −2.

Sum of zeroes = CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 16

Product of zeroes = 

CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 17

The value of t2 − 15 is zero when  or CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 18, i.e., when 

Therefore, the zeroes of t2 − 15 are CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 19 and.

Sum of zeroes =CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 20

Product of zeroes = 

CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 21

The value of 3x2 − x − 4 is zero when 3x − 4 = 0 or x + 1 = 0, i.e., when  or x = −1

Therefore, the zeroes of 3x2 − x − 4 are CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 22and −1.

Sum of zeroes = 

Product of zeroes CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 23

PAGE NO 33:

Question 2:

Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.

 CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 24 

CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 25  CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 26

ANSWER:

Let the polynomial be CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 27, and its zeroes be and CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 28.

Therefore, the quadratic polynomial is 4x2 − x − 4.

CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 24

Let the polynomial be , and its zeroes beCBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 30 and .

CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 31

Therefore, the quadratic polynomial is 3x2 − x + 1.

CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 32

Let the polynomial be , and its zeroes beCBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 30 and .

CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 34

Therefore, the quadratic polynomial is .

CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 35

Let the polynomial be , and its zeroes beCBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 30 and .

CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 37

Therefore, the quadratic polynomial is .

CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 38

Let the polynomial be , and its zeroes beCBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 30 and .

CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 40

Therefore, the quadratic polynomial is .

CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 41

Let the polynomial be .

CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 42

Therefore, the quadratic polynomial is.

PAGE NO 36:

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:

CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 43

ANSWER:

(i) 

CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 44

Therefore, CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 45, 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

CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 46

Therefore, the relationship between the zeroes and the coefficients is verified.

(ii) 

CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 47

Therefore, 2, 1, 1 are the zeroes of the given polynomial.

Comparing the given polynomial with CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 48, 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 CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 49

Multiplication of zeroes = 2 × 1 × 1 = 2 

Hence, the relationship between the zeroes and the coefficients is verified.

PAGE NO 36:

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:

Let the polynomial be CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 50and the zeroes be .

It is given that

CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 51

If a = 1, then b = −2, c = −7, d = 14

Hence, the polynomial is .

PAGE NO 36:

Question 3:

Obtain all other zeroes of CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 52, if two of its zeroes are .

ANSWER:

CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 53

Since the two zeroes are ,

CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 54is a factor of .

Therefore, we divide the given polynomial by CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 55.

We factorize CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 56

Therefore, its zero is given by x + 1 = 0

x = −1

As it has the term CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 57, therefore, there will be 2 zeroes at x = −1.

Hence, the zeroes of the given polynomial are, −1 and −1.

PAGE NO 36:

Question 4:

On dividing CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 58by a polynomial g(x), the quotient and remainder were − 2 and − 2x + 4, respectively. Find g(x).

ANSWER:

g(x) = ? (Divisor)

Quotient = (x − 2)

Remainder = (− 2x + 4)

Dividend = Divisor × Quotient + Remainder

CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 59

g(x) is the quotient when we divide byCBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 60

CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 61

PAGE NO 36:

Question 5:

Give examples of polynomial p(x), g(x), q(x) and r(x), which satisfy the division algorithm and

(i) deg p(x) = deg q(x)

(ii) deg q(x) = deg r(x)

(iii) deg r(x) = 0

ANSWER:

According to the division algorithm, if p(x) and g(x) are two polynomials with

g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that

p(x) = g(x) × q(x) + r(x),

where r(x) = 0 or degree of r(x) < degree of g(x)

Degree of a polynomial is the highest power of the variable in the polynomial.

(i) deg p(x) = deg q(x)

Degree of quotient will be equal to degree of dividend when divisor is constant ( i.e., when any polynomial is divided by a constant).

Let us assume the division of by 2.

Here, p(x) = CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 62

g(x) = 2

q(x) =  and r(x) = 0

Degree of p(x) and q(x) is the same i.e., 2.

Checking for division algorithm,

p(x) = g(x) × q(x) + r(x)

CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 62= 2()

CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 62

Thus, the division algorithm is satisfied.

(ii) deg q(x) = deg r(x)

Let us assume the division of x3 + x by x2,

Here, p(x) = x3 + x

g(x) = x2

q(x) = x and r(x) = x

Clearly, the degree of q(x) and r(x) is the same i.e., 1.

Checking for division algorithm,

p(x) = g(x) × q(x) + r(x)

x3 + x = (x) × x x

x3 + x = x3 + x

Thus, the division algorithm is satisfied.

(iii)deg r(x) = 0

Degree of remainder will be 0 when remainder comes to a constant.

Let us assume the division of x3 + 1by x2.

Here, p(x) = x3 + 1

g(x) = x2

q(x) = x and r(x) = 1

Clearly, the degree of r(x) is 0.

Checking for division algorithm,

p(x) = g(x) × q(x) + r(x)

x3 + 1 = (x) × x + 1

x3 + 1 = x3 + 1

Thus, the division algorithm is satisfied.

PAGE NO 37:

Question 3:

If the zeroes of polynomial  areCBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 65, find a and b.

ANSWER:

Zeroes are a − ba + a + b

Comparing the given polynomial with CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 66, we obtain

p = 1, q = −3, r = 1, t = 1

The zeroes are CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 67.

Hence, a = 1 and b = CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 68 or .

PAGE NO 37:

Question 4:

]It two zeroes of the polynomial CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 69 are, find other zeroes.

ANSWER:

Given that 2 +CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 70 and 2­­CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 70 are zeroes of the given polynomial.

Therefore, x2 + 4 ­­− 4x − 3

= x2 ­− 4x + 1 is a factor of the given polynomial

For finding the remaining zeroes of the given polynomial, we will find the quotient by dividing CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 69 by x2 ­− 4x + 1.

Clearly,CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 69 = 

It can be observed that CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 74is also a factor of the given polynomial.

And CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 75

Therefore, the value of the polynomial is also zero when or CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 76

Or x = 7 or −5

Hence, 7 and −5 are also zeroes of this polynomial.

PAGE NO 37:

Question 5:

If the polynomial  is divided by another polynomialCBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 77, the remainder comes out to be x + a, find k and a.

ANSWER:

By division algorithm,

Dividend = Divisor × Quotient + Remainder

Dividend − Remainder = Divisor × Quotient

 will be perfectly divisible by CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 77.

Let us divide  by CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 77

It can be observed thatCBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 80 will be 0.

Therefore, = 0 and CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 81= 0

For = 0,

2 k =10

And thus, k = 5

For CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials 81= 0

10 − a − 8 × 5 + 25 = 0

10 − a − 40 + 25 = 0

− 5 − a = 0

Therefore, a = −5

Hence, k = 5 and a = −5

Official Website – Click Here

Posts Related to CBSE Class 10 Maths NCERT Solutions

In this article we are provided the complete solutions for the CBSE Class 10 NCERT Solutions for Maths Chapter 2, Polynomials each and every topic with solutions are available in this website. Check once in this article and start your examination preparation immediately. Don,t forget to share in this article to our friends.

📢 Get Latest Exam Updates via E-mail ✉

Note : Submit your name, email, state and updates category below.
  • This field is for validation purposes and should be left unchanged.

Leave A Reply

Your email address will not be published.