Question

The constant term of the polynomial
$\left|\begin{array}{ccc}x+3& x& x+2\\ x& x+1& x-1\\ x+2& 2x& 3x+1\end{array}\right|$ is ________

• A. -1
• B. 1
• C. 0
• D. 2

Answer 1

Answer: (A)

$\left|\begin{array}{ccc}x+3& x& x+2\\ x& x+1& x-1\\ x+2& 2x& 3x+1\end{array}\right|=f(x)$
Applying $R_2\to R_2-R_1$ and $R_3\to R_3-R_1$
$f(x)=\left|\begin{array}{ccc}x+3& x& x+2\\ -3& 1& -3\\ -1& x& 2x-1\end{array}\right|$
Applying $C_1\to C_1-C_3\text{ and }{C}_2\to C_2-C_3$
$f(x)=\left|\begin{array}{ccc}1& -2& x+2\\ 0& 4& -3\\ -2x& 1-x& 2x-1\end{array}\right|$
Expand w.r.t. ' $C_1$ '
$f(x)=[4(2x-1)+3(1-x)]$
$+(-2x)[6-4(x+2)]$
$f(x)=[8x-4+3-3x]+[-2x][-4x-2]$
$f(x)=(5x-1)+\left(8x^2+4x\right)$
$f(x)=8x^2+9x-1$
Hence, the constant term of quadratic equation is $-1$