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If x^2+ax+1 is a factor of ax^3+ bx + c...

If `x^2+ax+1` is a factor of `ax^3+ bx + c`, then which of the following conditions are not valid: a. `a^2+c=0` b. `b-a=ac` c. `c^3+c+b^2=0` d. `2c+a=b`

Text Solution

Verified by Experts

The correct Answer is:
`b - a + a^(3) = 0, a^(2) + c = 0`
`x^(2) + ax + 1` must divide `ax^(3) + bx + c .` Now
`(ax^(3) +bx + c)/(x^(2) + ax + 1)= a (x - a) + ((b - a + a)^(3) x+c + a^(2))/(x^(2) + ax + 1)`
Ther remainder must be zero . Hence, `(b - a + a^(3)) x + c + a^(2) = 0 AA x in` R
So, `b - a + a^(3) = 0, a^(2) + c = 0`
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