Question

Total number of polynomials of the form $x^3+a{x}^2+bx+c$ that are divisible by $x^2+1$, where $a,b,c\in \{1,2,3,......10\}$ is equal to

• A. $90$
• B. $45$
• C. $5$
• D. $10$

Answer 1

The correct option is D $10$

Let $f\left(x\right)=x^3+a{x}^2+bx+c$

and $g\left(x\right)=x^2+1$

Dividing $f\left(x\right)$ by $g\left(x\right)$

$\Rightarrow f\left(x\right)=g\left(x\right)(x+a)+(b-1)x+(c-a)$

Now if $f\left(x\right)$ is divisible by $g\left(x\right)$

$\Rightarrow (b-1)x+(c-a)=0$

$\Rightarrow b-1=0$ and $c-a=0$

$\Rightarrow b=1$ 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.