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The polynomial P(x)=x^(3)+ax^(2)+bx+c ha...

The polynomial `P(x)=x^(3)+ax^(2)+bx+c` has the property that the mean of its roots, the product of its roots, and the sum of its coefficients are all equal. If the `y`-intercept of the graph of `y=P(x)` is `2`,
The value of `P(1)` is

A

`0`

B

`-1`

C

`2`

D

`-2`

Text Solution

Verified by Experts

The correct Answer is:
D
The `y`-intercept is at `x=0`, so we have `c=2`, meaning that the product of the roots is `-2`.
We know that `-a` is the sum of the roots.
The average of the roots is equal to the product, so the sum of the roots is `-6`, and `a=6`.
Finally, `1+a+b+c=2` as well, so we have `1+6+b+2=-2`
`impliesb=-11`
`:.P(x)=x^(3)+6x^(2)-11x+2`
`:.P(1)=1+6-11+2=-2`
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