What is the answer for NCERT Exersize 2.3 Q5?

Asked by Jaydip | 23rd Apr, 2017, 10:28: PM

Expert Answer:

Hi student,
 
This is the question:
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
Solution:

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) x q(x) + r(x), where r(x) = 0 or degree of r(x) < degree of g(x).

(i)    Degree of quotient will be equal to degree of dividend when divisor is constant.
Let us consider the division of 18x2 + 3x + 9  by 3.
Here, p(x) = 18x2 + 3x + 9  and g(x) = 3
q(x) = 6x2 + x + 3  and r(x) = 0
Here, degree of p(x) and q(x) is the same which is 2.


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

 

Thus, the division algorithm is satisfied.

(ii)    Let us consider the division of 2x4 + 2x by 2x3,
Here, p(x) = 2x4 + 2x and g(x) = 2x3
q(x) = x and r(x) = 2x
Clearly, the degree of q(x) and r(x) is the same which is 1.

Checking,
p(x) = g(x) x q(x) + r(x)
2x4 + 2x =  (2x3 ) x x  + 2x
2x4 + 2x = 2x4 + 2x
Thus, the division algorithm is satisfied.

(iii)    Degree of remainder will be 0 when remainder obtained on division is a constant.
Let us consider the division of 10x3 + 3 by 5x2.
Here, p(x) = 10x3 + 3 and g(x) = 5x2
q(x) = 2x and r(x) = 3
Clearly, the degree of r(x) is 0.

Checking:
p(x) = g(x) x q(x) + r(x)
10x3 + 3 = (5x2 ) x 2x  +  3
10x3 + 3 = 10x3 + 3
Thus, the division algorithm is satisfied.

Concept insight: In order to answer such type of questions, one should remember the division algorithm. Also, remember the condition on the remainder polynomial r(x). The polynomial r(x) is either 0 or its degree is strictly less than g(x). The answer may not be unique in all the cases because there can be multiple polynomials which satisfies  the given conditions.
 
Note: Dear student, all the solutions to NCERT are available on the site I am sharing the link : (Incase you need it for future reference.)
 
http://www.topperlearning.com/study/cbse/class-10/mathematics/text-book-solutions/ncert-mathematics-x/5/real-numbers/12/b101c2s3e9#show-question-225

Answered by Rebecca Fernandes | 27th Nov, 2017, 12:37: PM