Find the remainder when is divided by

(i)

(ii)

(iii)

(iv)

(v)

Remainder Theorem: If f(x) is a polynomial and it is divided by another polynomial g(x), the remainder of this division equals to the value f(a), where a is the solution of polynomial  g(x) = 0

for example: if x² + 2 is divided by x - 1, then to find remainder. 
 
put x - 1 = 0
x = 1 and now putting this value in x² + 1, we get 1 + 1 =2 as the remainder.

(i) f(x) = x³ + 3 x² + 3 x + 1

Now let g(x) = x + 1

So for finding remainder, put g(x) = 0

x + 1 = 0
x = -1
So f(- 1) will be the remainder when f(x) is divided by g(x)
f (- 1) = (- 1)³ + 3 (- 1)² + 3 (- 1) + 1
f (- 1) = -1 + 3 -3 + 1
f (- 1) = 0
Hence, Remainder = 0


(ii) f(x) = x³ + 3 x² + 3 x + 1

Now let g(x) = x - 1/2

So for finding remainder, put g(x) = 0
x - 1/2 = 0
x = 1/2
so, f(1/2) will be the remainder when f(x) is divided by g(x)

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





Hence, Remainder = 27/8


(iii) f(x) = x³ + 3 x² + 3 x + 1

Now let g(x) = x

So for finding remainder, put g(x) = 0
x = 0
So, f(0) will be the remainder when f(x) is divided by g(x)
f(0) = (0)³ + 3 (0)² + 3 (0) + 1
f(0) = 1

Hence, Remainder = 1

(iv) f(x) = x³ + 3 x² + 3 x + 1

Now let g(x) = x + Π

So for finding remainder, put g(x) = 0

x + Π = 0
x = - Π
So, f(- Π) will be the remainder when f(x) is divided by g(x)
f(- Π) = (- Π)³ + 3 (- Π)² + 3 (- Π) + 1
f(- Π) =  - Π³ + 3 Π² - 3Π + 1

- Π³ + 3 Π² - 3Π + 1 will be the remainder


(v) f(x) = x³ + 3 x² + 3 x + 1

Now let g(x) = 5 + 2x

So for finding the remainder, g(x) = 0

5 + 2x = 0
x = -5/2
So, f(-5/2) will be the remainder when f(x) is divided by g(x)









f(-5/2) = -27/8 will be the remainder .