Class 9 NCERT Solutions - Chapter 2 Polynomials - Exercise 2.3

Chapter 2 of the Class 9 NCERT Mathematics textbook, "Polynomials," delves into the concept of polynomials, their types, and various operations that can be performed on them. This chapter helps students understand the standard form of polynomials, the degree of a polynomial, and the significance of the zeroes of a polynomial. It also covers factorization methods and the relationship between the coefficients and the roots of a polynomial.

NCERT Solutions for Class 9 - Mathematics - Chapter 2 Polynomials - Exercise 2.3

This section provides detailed solutions to the problems in Exercise 2.3 of Chapter 2, "Polynomials," from the Class 9 NCERT Mathematics textbook. The exercise focuses on finding the zeroes of a polynomial, verifying relationships between the coefficients and roots, and applying these concepts to solve polynomial equations.

Question 1. Find the remainder when x³ + 3x² + 3x + 1 is divided by

(i) x + 1

Solution:

x + 1 = 0

x  = −1

Therefore remainder will be f(x):

f(−1) = (−1)³ + 3(−1)² + 3(−1) + 1

= −1 + 3 − 3 + 1

= 0

(ii) x – 1/2

Solution:

x - 1/2 = 0

x = 1/2

Therefore remainder will be f(x):

f(1/2) = (1/2)³ + 3(1/2)² + 3(1/2) + 1

= (1/8) + (3/4) + (3/2) + 1

= 27/8

(iii) x

Solution:

x = 0

Therefore remainder will be f(x):

f(0) = (0)³ + 3(0)² + 3(0) + 1

= 1

(iv) x + pi

Solution:

x + pi = 0

x = −pi

Therefore remainder will be f(x):

f(−pi) = (−pi)³ + 3(−pi)² + 3(−pi) + 1

= −pi³ + 3pi² − 3pi + 1

(v) 5 + 2x

Solution:

5 + 2x = 0

2x = −5

 x = -5/2

Therefore remainder will be f(x) :

f(-5/2) = (-5/2)³ + 3(-5/2)² + 3(-5/2) + 1 

= (-125/8) + (75/4) - (15/2) + 1

= -27/8

Question 2. Find the remainder when x³ − ax² + 6x − a is divided by x - a.

Solution:

Let f(x) = x³ − ax² + 6x − a

x − a = 0

∴ x = a

Therefore remainder will be f(x):

f(a) = (a)³ − a(a²) + 6(a) − a

= a³ − a³ + 6a − a 

= 5a

Question 3. Check whether 7 + 3x is a factor of 3x³ + 7x.

Solution:

7 + 3x = 0

3x = −7

 x = -7/3

Therefore remainder will be f(x):

f(-7/3) = 3(-7/3)³ + 7(-7/3) 

= - (343/9) + (-49/3)

= (-343- (49) * 3)/9

= (-343 - 147)/9

= - 490/9 ≠ 0

∴ 7 + 3x is not a factor of 3x³ + 7x

Summary

Exercise 2.3 of Chapter 2 "Polynomials" in the Class 9 NCERT Mathematics textbook focuses on finding the zeroes of polynomials and understanding the relationship between the coefficients and roots. The exercise includes problems that require students to find the zeroes of given polynomials, verify these zeroes, and explore their relationships with the coefficients. This exercise is crucial in developing a deeper understanding of how polynomials behave and how their roots influence the overall polynomial expression

Related Articles:

• Class 9 NCERT Solutions - Chapter 2 Polynomials - Exercise 2.2
• Class 9 NCERT Solutions - Chapter 2 Polynomials - Exercise 2.4