Question

Find the number of polynomials of the form $x^3+a{x}^2+bx+c$ that are divisible by $x^2+1$, where a, b, c $ϵ\{1,2,3,...,9,10\}$.

Answer 1

$x+a$
$x^2+1)\overline{x^3+a{x}^2+bx+c}$
$x^3+x$
$\overline{a{x}^2+(b-1)x+c}$
$a{x}^2+a$
$\overline{(b-1)x+c-a}$
Now, remainder $(b-1)x+c-a$ must be zero for any x. Then, $b-1=0$ and $c-a=0$
$\Rightarrow b=1$ and $c=a$
Now, c or a can be selected in 10 ways. Hence, the number of polynomials are 10.