If `x^2+ax+1` is a factor of `ax^3+ bx + c`, then which of the following conditions is valid: a. `a^2+c=0` b. `b-a=ac` c. `c^3+c+b^2=0` d. `2c+a=b`

Explanation

`:'ax^(3)+bx+c=(x^(2)+ax+1)Q(x)`
Let `Q(x)=Ax+B`
then `ax^(3)+bx+c=(x^(2)+ax+1)(Ax+B)`
On comparing coefficients of `x^(3),x^(2),x` and constant on both sides, we get
`a=a`……i
`0=B+aA`,……….ii
`b=aB+A`…………..iii
and `c=B`..........iv
From Eqs. i and iv we get
`A=a` and `B=c`
From Eqs. ii and iii `a^(2)+c=0` and `b=ac+a` are the required conditions.