Question

If $f(x),g(x)$and $h(x)$ are three polynomials of degree 2 and $\Delta \left(x\right)=\left|\begin{array}{ccc}f\left(x\right)& g\left(x\right)& h\left(x\right)\\ f^{\prime }\left(x\right)& g^{\prime }\left(x\right)& h^{\prime }\left(x\right)\\ f^{\prime \prime }\left(x\right)& g^{\prime \prime }\left(x\right)& h^{\prime \prime }\left(x\right)\end{array}\right|$ then $\Delta \left(x\right)$ is a polynomial of degree

• A. 2
• B. 3
• C. 0
• D. atmost 3

Answer 1

Answer: (C)

Since $f(x),g(x)$ and $h(x)$ are the polynomials of degree $2$, therefore
$f^{\prime \prime \prime }\left(x\right)=g^{\prime \prime \prime }\left(x\right)=h^{\prime \prime \prime }\left(x\right)=0$
Now, $\Delta \left(x\right)=\left|\begin{array}{ccc}f\left(x\right)& g\left(x\right)& h\left(x\right)\\ f^{\prime }\left(x\right)& g^{\prime }\left(x\right)& h^{\prime }\left(x\right)\\ f^{\prime \prime }\left(x\right)& g^{\prime \prime }\left(x\right)& h^{\prime \prime }\left(x\right)\end{array}\right|$
$\Rightarrow {\Delta }^{\prime }(x)=\left|\begin{array}{ccc}{f}^{\prime }\left(x\right)& g^{\prime }\left(x\right)& h^{\prime }\left(x\right)\\ f^{\prime }\left(x\right)& g^{\prime }\left(x\right)& h^{\prime }\left(x\right)\\ f^{\prime \prime }\left(x\right)& g^{\prime \prime }\left(x\right)& h^{\prime \prime }\left(x\right)\end{array}\right|$
$+\left|\begin{array}{ccc}f\left(x\right)& g\left(x\right)& h\left(x\right)\\ f^{\prime \prime }\left(x\right)& g^{\prime \prime }\left(x\right)& h^{\prime \prime }\left(x\right)\\ f^{\prime \prime }\left(x\right)& g^{\prime \prime }\left(x\right)& h^{\prime \prime }\left(x\right)\end{array}\right|$
$+\left|\begin{array}{ccc}{f}^{\prime \prime }\left(x\right)& g^{\prime \prime }\left(x\right)& h^{\prime \prime }\left(x\right)\\ f^{\prime }\left(x\right)& g^{\prime }\left(x\right)& h^{\prime }\left(x\right)\\ f^{\prime \prime \prime }\left(x\right)& g^{\prime \prime \prime }\left(x\right)& h^{\prime \prime \prime }\left(x\right)\end{array}\right|$
$\Rightarrow {\Delta }^{\prime }(x)=0+0+0=0$
$\Rightarrow \Delta (x)=$ constant.
Thus, $\Delta (x)$ is the polynomial of degree zero.