Answer :
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 .
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 .
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