NCERT Solutions for class 10th chapter - 2 Polynomials

                         Exercise 2.1

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

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Ans. (i) The graph does not meets x-axis at all. Hence, it does not have any zero.

(ii) Graph meets x-axis 1 time. It means this polynomial has 1 zero.

(iii) Graph meets x-axis 3 times. Therefore, it has 3 zeroes.

(iv) Graph meets x-axis 2 times. Therefore, it has 2 zeroes.

(v) Graph meets x-axis 4 times. It means it has 4 zeroes.

(vi) Graph meets x-axis 3 times. It means it has 3 zeroes.

                           Exercise 2.2

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

(i) 

(ii) 

(iii) 

(iv) 

(v) 

(vi) 

Ans. (i) 

Comparing given polynomial with general form ,

We get a = 1, b = -2 and c = -8

We have, 

= x(x−4)+2(x−4) = (x−4)(x+2)

Equating this equal to 0 will find values of 2 zeroes of this polynomial.

(x−4)(x+2) = 0

⇒ x = 4, −2 are two zeroes.

Sum of zeroes = 4 – 2 = 2 = 

= 

Product of zeroes = 4 × −2 = −8

= 

(ii) 

Here, a = 4, b = -4 and c = 1

We have, 

=

=2s(2s−1)−1(2s−1)
= (2s−1)(2s−1)

Equating this equal to 0 will find values of 2 zeroes of this polynomial.

⇒ (2s−1)(2s−1) = 0

⇒ s = 

Therefore, two zeroes of this polynomial are 

Sum of zeroes = = 1 = 

= 

Product of Zeroes = 

(iii) 

Here, a = 6, b = -7 and c = -3

We have, 

= 3x(2x−3)+1(2x−3) = (2x−3)(3x+1)

Equating this equal to 0 will find values of 2 zeroes of this polynomial.

⇒ (2x−3)(3x+1) = 0

⇒ x = 

Therefore, two zeroes of this polynomial are 

Sum of zeroes = 

Product of Zeroes = 

(iv) 

Here, a = 4, b = 8 and c = 0

Equating this equal to 0 will find values of 2 zeroes of this polynomial.

⇒ 4u(u+2) = 0

⇒ u = 0,−2

NCERT Solutions for Class 10 Maths Exercise 2.2

Therefore, two zeroes of this polynomial are 0, −2

Sum of zeroes = 0−2 = −2 = 

= 

Product of Zeroes = 0

= 

(v) 

Here, a = 1, b = 0 and c = -15

We have, ⇒ ⇒ t = 

Therefore, two zeroes of this polynomial are 

Sum of zeroes = 

Product of Zeroes = 

(vi) 

Here, a = 3, b = -1 and c = -4

We have, = 

= x(3x−4)+1(3x−4) = (3x−4)(x+1)

Equating this equal to 0 will find values of 2 zeroes of this polynomial.

⇒ (3x−4)(x+1) = 0

⇒ x = 

Therefore, two zeroes of this polynomial are 

Sum of zeroes = 

Product of Zeroes = 

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NCERT Solutions for Class 10 Maths Exercise 2.2

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

(i) , −1

(ii) , 13

(iii) 0, 

(iv) 1, 1

(v) 

(vi) 4, 1

Ans. (i) , −1

Let quadratic polynomial be 

Let α and β are two zeroes of above quadratic polynomial.

α+β =  = 

α × β = -1   = 

 Quadratic polynomial which satisfies above conditions = 

(ii) 

Let quadratic polynomial be 

Let α and β be two zeros of above quadratic polynomial.

α+β = = 

α × β =  which is equal to 

 

 Quadratic polynomial which satisfies above conditions = 

(iii) 0, 

Let quadratic polynomial be 

Let α and β be two zeros of above quadratic polynomial.

α+β = 0  = 

α  β =  = 

 

 Quadratic polynomial which satisfies above conditions 

(iv) 1, 1

Let quadratic polynomial be 

Let α and β be two zeros of above quadratic polynomial.

α+β = 1  = 

α β = 1  =

 

 Quadratic polynomial which satisfies above conditions = 

(v) 

Let quadratic polynomial be 

Let α and β be two zeros of above quadratic polynomial.

α+β = = 

α  β =  = 

 

 Quadratic polynomial which satisfies above conditions = 

(vi) 4, 1

Let quadratic polynomial be 

Let α and β be two zeros of above quadratic polynomial.

α+β = 4  = 

α × β = 1  = 

 

 Quadratic polynomial which satisfies above conditions 

                           Exercise 2.3

1. Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following.

(i), 

(ii), 

(iii), 

Ans. (i)

Therefore, quotient = x – 3 and Remainder = 7x – 9

(ii)

Therefore, quotient = and, Remainder = 8

(iii)

Therefore, quotient = and, Remainder = −5x + 10

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NCERT Solutions for Class 10 Maths Exercise 2.3

2. Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial.

(i) 

(ii) 

(iii) 

Ans. (i)

Remainder = 0

Hence first polynomial is a factor of second polynomial.

(ii)

 Remainder = 0

Hence first polynomial is a factor of second polynomial.

(iii)

 Remainder ≠0

Hence first polynomial is not factor of second polynomial.

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NCERT Solutions for Class 10 Maths Exercise 2.3

3. Obtain all other zeroes of if two of its zeroes are and .

Ans. Two zeroes of are and  which means that  is a factor of .

Applying Division Algorithm to find more factors we get:

We have 

⇒ 

= ()

= ()3

= 3()

= 3()(x+1)(x+1)

Therefore, other two zeroes of are −1 and −1.

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NCERT Solutions for Class 10 Maths Exercise 2.3

4. On dividing by a polynomial g(x), the quotient and remainder were (x-2) and (-2x+4) respectively. Find g(x).

Ans. Let, q(x) = (x – 2) and r(x) = (–2x+4)

According to Polynomial Division Algorithm, we have

p(x) = g(x).q(x) + r(x)

⇒ = g(x).(x−2)−2x+4

⇒ −4 = g(x).(x−2)

⇒ = g(x).(x−2)

⇒ g(x) = 

So, Dividing by (x−2), we get

Therefore, we have g(x) = 

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NCERT Solutions for Class 10 Maths Exercise 2.3

5. Give examples of polynomials 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

Ans. (i) Let , g(x) = 3

So, we can see in this example that deg p(x) = deg q(x) = 2

(ii) Let and 

We can see in this example that deg q(x) = deg r(x) = 1

(iii) Let, g(x) = x+3

We can see in this example that deg r(x) = 0


                Exercise 2.4(optional)

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:

(i) 

(ii) 

Ans. (i) Comparing the given polynomial with , we get

 and 

=  =  = 0

=  = 0

=  =  = 0

 and  are the zeroes of 

Now, 

= 

And 

= 

=  = 

And  = 

(ii) Comparing the given polynomial with , we get

 and 

=  = 0

=  = 0

 and  are the zeroes of 

Now, 

= 

And  = 

=  = 

And  = 

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NCERT Solutions for Class 10 Maths Exercise 2.4

2. Find a cubic polynomial with the sum of the product of its zeroes taken two at a time and the product of its zeroes are  respectively.

Ans. Let the cubic polynomial be  and its zeroes be  and 

Then  = 2 = and  = 

And  = 

Here,  and 

Hence, cubic polynomial will be 

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NCERT Solutions for Class 10 Maths Exercise 2.4

3. If the zeroes of the polynomial  are  find  and 

Ans. Since  are the zeroes of the polynomial 

 = 

 

And 

= 

 

 

  

 

 

Hence  and .

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NCERT Solutions for Class 10 Maths Exercise 2.4

4. If the two zeroes of the polynomial  are  find other zeroes.

Ans. Since  are two zeroes of the polynomial 

Let 

Squaring both sides, 

 

Now we divide  by  to obtain other zeroes.

= 

= 

= 

= 

 and  are the other factors of 

 and 7 are other zeroes of the given polynomial.

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NCERT Solutions for Class 10 Maths Exercise 2.4

5. If the polynomial  is divided by another polynomial  the remainder comes out to be  find  and 

Ans. Let us divide  by .

 Remainder = 

On comparing this remainder with given remainder, i.e. 

 

And