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Let P(x) be a polynomial of degree at mo...

Let `P(x)` be a polynomial of degree at most `5` which leaves remainders `-1` and `1` upon divison by `(x-1)^(3)` and `(x+1)^(3)`, respectively.
The sum of pairwise product of all roots ( real and complex) of `P(x)=0` is

A

`1`

B

`3`

C

`5`

D

`2`

Text Solution

Verified by Experts

The correct Answer is:
A
`P(x)+1=0` has a thrice repeated root at `x=1`
`P'(x)=0` has a twice repeated root at `x=1`.
Similarly `P'(x)=0` has a twice repeated root at `x=-1`
`implies P'(x)` is divisible by `(x-1)^(2)(x+1)^(2)`
`:'P'(x)` is of degree at most `4`.
`implies P'(x)=alpha(x-1)^(2)(x+1)^(2)`
`P(x)=alpha((x^(5))/(5)-(2)/(3)x^(3)+x)+c`
Now `P(1)=-1` and `P(-1)=1`
`implies alpha=-(15)/(8)` and `c=0`
`:. P(x)=-(3)/(8)x^(5)+(5)/(4)x^(3)-(15)/(8)x`
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