Question

If $f\left(x\right),g\left(x\right)$ and $h\left(x\right)$ are three polynomial 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 $\le 2$, if true enter 1 else 0.

Answer 1

Let $f\left(x\right)=a_0{x}^2+a_{1}x+a_{2}g\left(x\right)=b_0{x}^2+b_{1}x+b_{2}h\left(x\right)=c_0{x}^2+c_{1}x+c_2$

Then, $\Delta \left(x\right)=\left|\begin{array}{ccc}f\left(x\right)& g\left(x\right)& h\left(x\right)\\ 2a_{0}x+a_1& 2b_{0}x+b_1& 2c_{0}x+c_1\\ 2a_0& 2b_0& 2c_0\end{array}\right|$

$=x\left|\begin{array}{ccc}f\left(x\right)& g\left(x\right)& h\left(x\right)\\ 2a_0& 2b_0& 2c_0\\ 2a_0& 2b_0& 2c_0\end{array}\right|+\left|\begin{array}{ccc}f\left(x\right)& g\left(x\right)& h\left(x\right)\\ a_1& b_1& c_1\\ 2a_0& 2b_0& 2c_0\end{array}\right|$

$=0+2\left|\begin{array}{ccc}f\left(x\right)& g\left(x\right)& h\left(x\right)\\ a_1& b_1& c_1\\ a_0& b_0& c_0\end{array}\right|$

$=2[(b_1{c}_0-b_0{c}_1)f\left(x\right)-(a_1{c}_0-a_0{c}_1)g\left(x\right)+h\left(x\right)(a_1{b}_0-a_0{b}_1)]$

Hence degree of $\Delta \left(x\right)\le 2$