Question

If $f\left(x\right),g\left(x\right)$ and $h\left(x\right)$ are three polynomials of degree $2$ and $\Delta \left(x\right)\left|\begin{array}{c}f\left(x\right)\\ f^{\prime }\left(x\right)\\ f"\left(x\right)\end{array}\begin{array}{c}g\left(x\right)\\ g^{\prime }\left(x\right)\\ g"\left(x\right)\end{array}\begin{array}{c}h\left(x\right)\\ h^{\prime }\left(x\right)\\ g"\left(x\right)\end{array}\right|$ then polynomial of degree (whenever defined)

• A. $2$
• B. $3$
• C. $0$
• D. $1$

Answer 1

The correct option is C $0$

$△\left(x\right)=\left|\begin{array}{ccc}f\left(x\right)& g\left(x\right)& h\left(x\right)\\ f^{{}^{\prime }}\left(x\right)& g^{{}^{\prime }}\left(x\right)& h^{{}^{\prime }}\left(x\right)\\ f^{{}^{\prime \prime }}\left(x\right)& g^{{}^{\prime \prime }}\left(x\right)& h^{{}^{\prime \prime }}\left(x\right)\end{array}\right|$

$\Rightarrow f\left(x\right)[g^{{}^{\prime }}\left(x\right).h^{{}^{\prime \prime }}\left(x\right)-h^{{}^{\prime }}\left(x\right)g^{{}^{\prime \prime }}\left(x\right)]-g\left(x\right)[f^{{}^{\prime }}\left(x\right)h^{{}^{\prime \prime }}\left(x\right)-h^{{}^{\prime }}\left(x\right)f^{{}^{\prime \prime }}\left(x\right)]+h\left(x\right)[f^{{}^{\prime }}\left(x\right)g^{{}^{\prime \prime }}\left(x\right)-g^{{}^{\prime }}\left(x\right)f^{{}^{\prime \prime }}\left(x\right)]$

$\because f\left(x\right),g\left(x\right),h\left(x\right)$ are polynomial of degree $2$

$\therefore f^{{}^{\prime }}\left(x\right),g^{{}^{\prime }}\left(x\right),h^{{}^{\prime }}\left(x\right)$ are of degree $1$

$\therefore f^{{}^{\prime \prime }}\left(x\right),g^{{}^{\prime \prime }}\left(x\right),h^{{}^{\prime \prime }}\left(x\right)$ are of degree $0$

$\therefore △\left(x\right)$ is polynomial of degree $3$