# What is the answer for NCERT Exersize 2.3 Q5?

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

### 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 18x^{2} + 3x + 9 by 3.

Here, p(x) = 18x^{2} + 3x + 9 and g(x) = 3

q(x) = 6x^{2} + 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 2x^{4} + 2x by 2x^{3},

Here, p(x) = 2x^{4} + 2x and g(x) = 2x^{3}

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)

2x^{4} + 2x = (2x^{3} ) x x + 2x

2x^{4} + 2x = 2x^{4} + 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 10x^{3} + 3 by 5x^{2}.

Here, p(x) = 10x^{3} + 3 and g(x) = 5x^{2}

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)

10x^{3} + 3 = (5x^{2} ) x 2x + 3

10x^{3} + 3 = 10x^{3} + 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

**This is the question:**

(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 18x^{2} + 3x + 9 by 3.

Here, p(x) = 18x^{2} + 3x + 9 and g(x) = 3

q(x) = 6x^{2} + 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 2x

^{4}+ 2x by 2x

^{3},

Here, p(x) = 2x

^{4}+ 2x and g(x) = 2x

^{3}

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)

2x

^{4}+ 2x = (2x

^{3}) x x + 2x

2x

^{4}+ 2x = 2x

^{4}+ 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 10x

^{3}+ 3 by 5x

^{2}.

Here, p(x) = 10x

^{3}+ 3 and g(x) = 5x

^{2}

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)

10x

^{3}+ 3 = (5x

^{2}) x 2x + 3

10x

^{3}+ 3 = 10x

^{3}+ 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.)

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

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