Question

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

• A. a degree 1 polynomial
• B. constant polynomial
• C. a quadratic polynoial
• D. none of these

Answer 1

Answer: (B)

constant polynomial

Let, $f\left(x\right)=a_1{x}^2+b_{1}x+c_1$
$g\left(x\right)=a_2{x}^2+b_{2}x+c_2$
$h\left(x\right)=a_3{x}^2+b_{3}x+c_3$
Applying, $R_2=R_2-\frac{a_2}{a_1}{R}_1$ and $R_3=R_3-\frac{a_3}{a_1}{R}_1$,

$\Rightarrow \left|\begin{array}{ccc}{a}_1{x}^2+b_{1}x+c_1& a_2{x}^2-a_2{x}^2+b_{2}x-\frac{a_2{b}_{1}x}{a_1}+c_2-\frac{a_2{c}_1}{a_1}& a_3{x}^2-a_3{x}^2+b_{3}x-\frac{a_3{b}_{1}x}{a_1}+c_3-\frac{a_3{c}_1}{a_1}\\ 2a_{1}x+b_1& 2a_{2}x+b_2-2a_{2}x-\frac{b_1{a}_2}{a_1}& 2a_{3}x+b_3-2a_{3}x-\frac{b_1{a}_3}{a_1}\\ 2a_1& 0& 0\end{array}\right|$

Expanding by third row,
$2a_1(b_3{c}_2-\frac{a_2{c}_1{b}_3}{a_1}-\frac{b_1{a}_3{c}_2}{a_1}-b_2{c}_3+\frac{a_3{c}_1{b}_2}{a_1}+\frac{c_3{b}_1{a}_2}{a_1})$
which is a constant polynomial.