If `a^2+b^2+c^2=-2a n df(x)=` `|a+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. 0
B. 1
C. 2
D. 3

Answer

C

Explanation

Operating `C_(1) toC_(1)+C_(2) +C_(3)` we get
`f(x) = |{:(1+2x+(a^(2)+b^(2)+c^(2))x,,(a+b^(2))x,,(1+c^(2))x),(1+2x+(a^(2)+b^(2)+c^(2))x,,1+b^(2)x,,(1+c^(2))x),(1+2x+(a^(2)+b^(2)+c^(2))x,,(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),(1,,(1+b^(2))x,,1+c^(2)x):}|" " [ :' a^(2) +b^(2) +c^(2) =-2]`
`=|{:(1,,(1+b^(2))x,,(1+c^(2))x),(0,,1-x,,0),(0,,0,,1-x):}|" "underset(" and " R_(3) to R_(3) -R_(1))(" Operating " R_(2) to R_(2) -R_(1))`
`=(1)[(1-x)^(2)-0]`
`=(1-x)^(2)`
which is a polynomial of degree2