if `x^2+x+1` is a factor of `ax^3+bx^2+cx+d` then the real root of `ax^3+bx^2+cx+d=0 ` is : (a) `-d/a ` (B) `d/a ` (C) `a/b` (D)none of these

Options

A. `-d/a`
B. `d/a`
C. `a/d`
D. none of these

Answer

A

Explanation

We know that `x^(2)+x+1` is factor of `ax^(3)+bx^(2)+cx+d`
Hence roots of `x^(2)+x+1=0` are also roots of
`ax^(3)+bx^(2)+cx+d=0`. Since `omega` and `omega^(2)`
(where `omega=-1/2+(3i)/2`) are two complex roots of `x^(2)+x+1=0`
Therefore `omega` and `omega^(2)` are two complex roots fo `ax^(3)+bx^(2)+cx+d=0`
We know that a cubic equation has atleast one real root. Let real root be `alpha`. Then
`alpha. omega. omega^(2)=-d/aimpliesalpha=-d/a`