If `a^2+b^2+c^2=-2a n df(x)=` `|1+a^2x(1+b^2)x(1+c^2)x(1+a^2)x1+b^2x(1+c^2)x(1+a^2)x(1+b^2)x1+c^2x|` , then `f(x)` is a polynomial of degree `0` b. `1` c. `2` d. `3`

Options

A. 2
B. 3
C. 0
D. 1

Answer

A

Explanation

Applying `C_(1) rarr C_(1) + C_(2) + C_(3)`, we have
`f(x) = |(1 + x(a^(2) + b^(2) + c^(2) + 2),(1 +b^(2)) x,(1 + c^(2)) x),(1 + x (a^(2) + b^(2) + c^(2) + 2),1 + b^(2) x,(1 + c^(2)) x),(1 + x (a^(2) + b^(2) + c^(2) ++ 2),(1 + b^(2)) x,1 + c^(2) x)|`
`rArr f(x) =|(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)| " " [ :' a^(2) + b^(2) + c^(2) + 2 = 0]`
`rArr f(x)= |(1,(1 + b^(2))x,(1 + c^(2)) x),(0,1 -x,0),(0,0,1 - x)| [("Applying " R_(2) rarr R_(2) - R_(1)),(R_(3) rarr R_(3) - R_(1))]`
`rArr f(x) = (1 -x)^(2)`, which is a polyonmial of degree 2