If `a ,b ,c in R ,` then find the number of real roots of the equation `=|x c-b-c x a b-a x|=0`

Explanation

from the symmetry of the determinant it is easier to expand by sarrus rule.
`Delta =x^(3) +abc -abc + (b^(2)x +a^(2) + a^(2)x+ c^(2)x )=0`
` " or " x^(3) + x( a^(2) + b^(2) + c^(2))=0`
`rArr x=0 " or " x^(2) = -(a^(2) +b^(2) + c^(2))`
` rArr x=0 " or " x= +- i sqrt( a^(2) + b^(2) +c^(2))`