p(x) be a polynomial of degree at most 5 which leaves remainder - 1 and 1 upon division by `(x-1)^3` and `(x+1)^3`respectively, the number of real roots of `P(x) = 0` is

Options

A. `1`
B. `3`
C. `5`
D. `2`

Answer

A

Explanation

`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`