If a2+b2+c2=−2 and f(x)=1+a2x1+b2x1+c2x1+a2x1+b2x1+c2x1+a2x1+b2x1+c2x then f(x) is a polynomial of degree

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

1. A

0

2. B

1

3. C

2

4. D

3

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### Solution:

Operating ${\mathrm{C}}_{1}\to {\mathrm{C}}_{1}+{\mathrm{C}}_{2}+{\mathrm{C}}_{3}$, we get

$\mathrm{f}\left(\mathrm{x}\right)=\left|\begin{array}{ccc}1+2\mathrm{x}+\left({\mathrm{a}}^{2}+{\mathrm{b}}^{2}+{\mathrm{c}}^{2}\right)\mathrm{x}& \left(1+{\mathrm{b}}^{2}\right)\mathrm{x}& \left(1+{\mathrm{c}}^{2}\right)\mathrm{x}\\ 1+2\mathrm{x}+\left({\mathrm{a}}^{2}+{\mathrm{b}}^{2}+{\mathrm{c}}^{2}\right)\mathrm{x}& 1+{\mathrm{b}}^{2}\mathrm{x}& \left(1+{\mathrm{c}}^{2}\right)\mathrm{x}\\ 1+2\mathrm{x}+\left({\mathrm{a}}^{2}+{\mathrm{b}}^{2}+{\mathrm{c}}^{2}\right)\mathrm{x}& \left(1+{\mathrm{b}}^{2}\right)\mathrm{x}& 1+{\mathrm{c}}^{2}\mathrm{x}\end{array}\right|$

$\begin{array}{l}=\left(1\right)\left[\left(1-\mathrm{x}{\right)}^{2}-0\right]\\ =\left(1-\mathrm{x}{\right)}^{2}\end{array}$

which is a polynomial of degree 2.

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