Question

If $f(x),g(x)$ and $h(x)$ are three polynomials of degree 3 then $\varphi (x)=\left|\begin{array}{ccc}{f}^{\prime }(x)& g^{\prime }(x)& h^{\prime }(x)\\ f^{\prime \prime }(x)& g^{\prime \prime }(x)& h^{\prime \prime }(x)\\ f^{\prime \prime \prime }(x)& g^{\prime \prime \prime }(x)& h^{\prime \prime \prime }(x)\end{array}\right|$is a polynomial of degree

• A. 3
• B. 4
• C. 5
• D. none of these

Answer 1

Answer: (D)

We have
${\varphi }^{\prime }(x)=\left|\begin{array}{ccc}{f}^{\prime \prime }(x)& g^{\prime \prime }(x)& h^{\prime \prime }(x)\\ f^{\prime \prime }(x)& g^{\prime \prime }(x)& h^{\prime \prime }(x)\\ f^{\prime \prime \prime }(x)& g^{\prime \prime \prime }(x)& h^{\prime \prime \prime }(x)\end{array}\right|+\left|\begin{array}{ccc}{f}^{\prime }(x)& g^{\prime }(x)& h^{\prime }(x)\\ f^{\prime \prime \prime }(x)& g^{\prime \prime \prime }(x)& h^{\prime \prime \prime }(x)\\ f^{\prime \prime \prime }(x)& g^{\prime \prime \prime }(x)& h^{\prime \prime \prime }(x)\end{array}\right|+\left|\begin{array}{ccc}{f}^{\prime }(x)& g^{\prime }(x)& h^{\prime }(x)\\ f^{\prime \prime }(x)& g^{\prime \prime }(x)& h^{\prime \prime }(x)\\ f^{iv}(x)& g^{iv}(x)& h^v(x)\end{array}\right|$
$=0+0+\left|\begin{array}{ccc}{f}^{\prime }(x)& g^{\prime }(x)& h^{\prime }(x)\\ <br/>f^{\prime \prime }(x)& g^{\prime \prime }(x)& h^{\prime \prime }(x)\\ 0& 0& 0<br/>\end{array}\right|=0$
since $f,g,h$ are polynomials of degree $3,f^{iv}(x)=g^{iv}(x)=$ $h^{iv}(x)=0$
$\Rightarrow \varphi (x)$ must be a constant.