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If x^(2)+x+1 is a factor of a x^(3)+b x^...

If `x^(2)+x+1` is a factor of a `x^(3)+b x^(2)+c x+d`, then the real root of `a x^(3)+b x^(2)+c x+d=0` is

A

`-d/a`

B

`d/a`

C

`a/d`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
A
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`
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