Question

If $a^2+b^2+c^2=-2$ and $f(x)=\left|\begin{array}{ccc}1+a^{2}x& (1+b^2)x& (1+c^2)x\\ (1+a^2)x& 1+b^{2}x& (1+c^2)x\\ (1+a^2)x& (1+b^2)x& 1+c^{2}x\end{array}\right|$ then f(x) is a polynomial of degree

• A. 2
• B. 3
• C. 0
• D. 1

Answer 1

Answer: (A)

Operate $C_1+C_2+C_3$, we get
$f(x)=\left|\begin{array}{ccc}1+a^{2}x+b^{2}x+x{c}^{2}x& (1+b^2)x& (1+c^2)x\\ x+a^{2}x+1+b^{2}x+x+c^{2}x& 1+b^{2}x& (1+c^2)x\\ x+a^{2}x+x+b^{2}x+1+c^{2}x& (1+b^2)x& 1+c^{2}x\end{array}\right|$
= $(1+2x+(a^2+b^2+c2)x)$
$\left|\begin{array}{ccc}1& (1+b^2)x& (1+c^2)x\\ 1& 1+b^{2}x& (1+c^2)x\\ 1& (1+b^2)x& 1+c^{2}x\end{array}\right|$
$\left|\begin{array}{ccc}1& (1+b^2)x& (1+c^2)x\\ 0& 1-x& 0\\ 0& 0& 1-x\end{array}\right|$
[$\because a^2+b^2+c^2=-2\therefore 1+2x+(c{a}^2+b^2+c^2)x=1+2x-2x=1$]
= $(1-x{)}^2$ which is a polynomial of degree 2.