The number of polynomial of the form x3 + ax2 + bx + c which are of divisible by x2 + 1 and where area, b and c belongs to {1, 2 … 10} isClass:                9

# The number of polynomial of the form x3 + ax2 + bx + c which are of divisible by x2 + 1 and where area, b and c belongs to {1, 2 … 10} isClass:                9

1. A
10
2. B
100
3. C
70
4. D
80

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### Solution:

We have f(x) = x3 + ax2 + bx + c and g(x) = x2 + 1
Dividing f(x) by g(x)
f(x) = g(x)(x + a) + (b 1)x + (c a) and
if f(x) is divisible by g(x):
(b 1)x + (c a) = 0
b 1 = 0 and c a = 0
b = 1 and c = a
As a and c related to {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
then a and c can be choose in 10 ways
10 possible polynomial are formed.
Hence option (1) is correct.

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