Question

$\left|\begin{array}{ccc}{x}^2+1& x+2& 2\\ 2x^2+1& x-1& 1\\ 3x^2+1& 4x-1& 1\end{array}\right|=A{x}^3+B{x}^2+Cx+D$ then the value of $C$ is

Answer 1

$\left|\begin{array}{ccc}{x}^2+1& x+2& 2\\ 2x^2+1& x-1& 1\\ 3x^2+1& 4x-1& 1\end{array}\right|$

$R_2\to R_2-R_1;R_3\to R_3-R_2$

$\left|\begin{array}{ccc}{x}^2+1& x+2& 2\\ x^2& -3& -1\\ x^2& 3x& 0\end{array}\right|$

$=(x^2+1)\left(3x\right)-(x+2)\left(x^2\right)+2(3x^3+3x^2)$

$=3x^3+3x-x^3-2x^2+6x^3+6x^2$

$=8x^3+4x^2+3x=A{x}^3+B{x}^2+Cx+D$

$\therefore C=3$