__Exercise 2.3 Page: 40__

__Exercise 2.3 Page: 40__

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

**(i) x+1**

Solution:

x+1= 0

⇒x = −1

∴Remainder:

p(−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

∴Remainder:

p(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

∴Remainder:

p(0) = (0)^{3}+3(0)^{2}+3(0)+1

= 1

**(iv) x+π**

Solution:

x+π = 0

⇒ x = −π

∴Remainder:

p(0) = (−π)^{3 }+3(−π)^{2}+3(−π)+1

= −π^{3}+3π^{2}−3π+1

**(v) 5+2x**

Solution:

5+2x=0

⇒ 2x = −5

⇒ x = -5/2

∴Remainder:

(-5/2)^{3}+3(-5/2)^{2}+3(-5/2)+1 = (-125/8)+(75/4)-(15/2)+1

= -27/8

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

Solution:

Let p(x) = x^{3}−ax^{2}+6x−a

x−a = 0

∴x = a

Remainder:

p(a) = (a)^{3}−a(a^{2})+6(a)−a

= a^{3}−a^{3}+6a−a = 5a

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

Solution:

7+3x = 0

⇒ 3x = −7

⇒ x = -7/3

∴Remainder:

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 3x^{3}+7x

Class 9th Maths Polynomial, Class 9th Maths Polynomial