Question

Let `f(x)=x^(3)+x+1`, let `p(x)` be a cubic polynomial such that the roots of `p(x)=0` are the squares of the roots of `f(x)=0` , then
A. `p(1)=3`
B. the value of `P(n)`, `n in N` is odd
C. Sum of all roots of `p(x)=0` is `-2`
D. Sum of all product of roots taken two at a time is `1`

Answer 1

Correct Answer - A::B::C::D
`(a,b,c,d)` Let `x=alpha^(2)`, `alpha=sqrt(x)`, put this value in `x^(3)+x+1=0`
We get `xsqrt(x)+sqrt(x)+1=0`
`sqrt(x)(x+1)=-1`
`impliesx(x+1)^(2)=1`
`impliesx^(3)+2x^(2)+x-1=0`
`impliesp(x)=x^(3)+2x^(2)+x-1`
`impliesp(1)=3`
Also `p(1)` is odd when `n` is odd or even.