consider the fourth -degree polynomial equation
`|{:(a_(1)+b_(1)x,,a_(1)x^(2)+b_(1),,c_(1)),(a_(2)+b_(2)x^(2),,a_(2)x^(2)+b_(2),,c_(2)),(a_(3)+b_(3)x^(2),,a_(3)x^(2)+b_(3),,c_(3)):}|=0`
Without expanding the determinant find all the roots of the equation.

Explanation

`|{:(a_(1)+b_(1)x,,a_(1)x^(2)+b_(1),,c_(1)),(a_(2)+b_(2)x^(2),,a_(2)x^(2)+b_(2),,c_(2)),(a_(3)+b_(3)x^(2),,a_(3)x^(2)+b_(3),,c_(3)):}|=0 ("As " C_(1) " and " C_(2) " are indentical")`
So `x= +-1` are roots of the given equation . From Sarrus rule we observe that the degree of equation is 4.