Question

Total number of polynomials of the form $x^3+a{x}^2+bx+c,$ that are divisible by $x^2+1,$ where $a,b,c\in \{1,2,3,\dots ,9,10\}$ is

• A. $8$
• B. $10$
• C. $15$
• D. $30$

Answer 1

The correct option is B $10$

Let $p\left(x\right)=x^3+a{x}^2+bx+c$
$p\left(x\right)$ is divisible by $x^2+1=(x-i)(x+i)$
Hence, $p\left(i\right)=0$ and $p(-i)=0$
i.e., $i^3+a{i}^2+bi+c=0$
and $(-i{)}^3+a(-i{)}^2+b(-i)+c=0$
$\Rightarrow (c-a)+(b-1)i=0$
and $(c-a)+(1-b)i=0$
$\therefore a=c,b=1$
$\therefore$ Total number of such polynomials$={}^{10}{C}_1=10$.