These NCERT Solutions for Class 9 Maths Chapter 2 Polynomials Ex 2.3 Questions and Answers are prepared by our highly skilled subject experts.

## NCERT Solutions for Class 9 Maths Chapter 2 Polynomials Exercise 2.3

Question 1.

Find the remainder when x^{3} + 3x^{2} + 3x + 1 is divided by

(i) x + 1

(ii) x – \(\frac{1}{2}\)

(iii) x

(iv) x + p

(v) 5 + 2x

Solution:

(i) By long division, we have:

Here, the remainder is 0.

(ii) By long division, we have:

Here, the remainder is \(\frac{27}{8}\)

(iii) By long division, we have

Here, the remainder is 1

(iv) we have given

x^{3} + 3x^{2} + 3x + 1 ÷ x + π

or, x^{3} + 3x^{2} + 3x + 1 ÷ x + \(\frac{22}{7}\) (∵ π = \(\frac{22}{7}\))

By long division,

Here, the remainder is \(-\frac{3669}{49}\)

(v) We have given

x^{3} + 3x^{2} + 3x + 1 ÷ 5 + 2x

or, x^{3} + 3x^{2} + 3x + 1 ÷ 2x + 5

By long division, we have

Here, remainder is \(-\frac{27}{8}\)

Question 2.

Find the remainder when x^{3} – ax^{2} + 6x – a is divided by x – a.

Solution:

Here, p(x) = x^{3} – ax^{2} + 6x – a, and the zero of x – a is a

So, p(a) = a^{3} – a(a)^{2} + 6(a) – a

= a^{3} – a^{3} + 6a – a

= 5a

So, 5a is the remainder when x^{3} – ax + 6x – a is divided by x – a.

Question 3.

Check whether 7 + 3x is a factor of 3x^{3} + 7x.

Solution:

We know that 7 + 3x is a factor of 3x^{3} + 7x only if 7 + 3x divides 3x^{3} + 7x leaving no remainder.

Here, p(x) = 3x^{3} + 7x and the zero of 7 + 3x is \(-\frac{7}{3}\). so,

Here, the remainder is \(\frac{-490}{9}\).

Therefore 7 + 3x is not the factor of 3x^{3} + 7x.