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
  



SolutionSince 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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