CBSE 10th Math Chapter 2, Polynomials - Solutions of NCERT Mathematics Textbook Exercise 2.3

Class X, Mathematics, NCERT (CBSE) Solutions

Chapter 2, Polynomials (Division Algorithm for Polynomials

Solutions of NCERT Math Textbook Exercise 2.3

(Page 36)

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

(i) p(x) = x³ – 3x² + 5x – 3, g(x) = x² – 2   

(ii) p(x) = x⁴ – 3x² + 4x + 5, g(x) = x² + 1 – x

(iii) p(x) = x⁴ – 5x + 6, g(x) = 2 – x²

Solution:

(i) p(x) = x³ – 3x² + 5x – 3, g(x) = x² – 2  

 

(ii) p(x) = x⁴ – 3x² + 4x + 5, g(x) = x² + 1 – x

(iii) Do it yourself by taking hint from the above solutions.

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

(i) t² – 3, 2t⁴ + 3t³ – 2t² – 9t – 12

(ii) x² + 3x + 1, 3x⁴ + 5x³ – 7x² + 2x + 2

(iii) x³ – 3x + 1, x⁵ – 4x³ + x2 + 3x + 1  

Solution:

(i) Try to solve this problem yourself after going through the solved examples given in the NCERT mathematics textbook itself.

Solution: (ii) x² + 3x + 1, 3x⁴ + 5x³ – 7x² + 2x + 2

Solution: (iii) x³ – 3x + 1, x⁵ – 4x³ + x2 + 3x + 1  

Q 3: Obtain all other zeroes of 3x⁴ + 6x³ – 2x² – 10x – 5, if two of its zeroes are

  

Solution: Since two zeroes are 

Q 4: On dividing x³ – 3x² + x + 2 by a polynomial g(x), the quotient and the remainder were x – 2 and -2x + 4, respectively. Find g(x).

Solution:

By the division algorithm, we have

f(x) = g(x) x q(x) + r(x)

Or, g(x) x q(x) = f(x) – r(x)

Or, g(x) (x – 2) = x³ – 3x² + x + 2 – (-2x + 4)

Or, g(x) (x – 2) = x³ – 3x² + 3x – 2

Thus, g(x) is a factor of x³ – 3x² + 3x – 2 other than the factor (x – 2). Hence, to get g(x) we will divide

(x³ – 3x² + 3x – 2) by (x – 2),

    

Q 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   

Solution:

(i) deg p(x) = deg q(x)

Let us assume the division of 6x² + 2x + 2 by 2

Here, p(x) = 6x² + 2x + 2

g(x) = 2

q(x) = 3x² + x + 1

r(x) = 0

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

Checking for division algorithm,

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

Or, 6x² + 2x + 2 = 2x (3x² + x + 1)

Hence, division algorithm is satisfied.

Solution: (ii) and (iii) to be loaded soon.

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