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Question

Total number of polynomials of the form x3+ax2+bx+c that are divisible by x2+1, where a,b,c{1,2,3,......10} is equal to

A
90
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B
45
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C
5
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D
10
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Solution

The correct option is D 10
Let f(x)=x3+ax2+bx+c

and g(x)=x2+1

Dividing f(x) by g(x)

f(x)=g(x)(x+a)+(b1)x+(ca)

Now if f(x) is divisible by g(x)

(b1)x+(ca)=0

b1=0 and ca=0

b=1 and c=a

As a and c belongs to {1,2,3,4,5,6,7,8,9,10}

So, a and c can be chosen in 10 ways

So, 10 such polynomials are possible .

So, option D is correct.

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