Find the values of `p\ a n d\ q` so that `x^4+p x^3+2x^2-3x+q` is divisible by `(x^2-1)`

Explanation

1409717 Let `f(x)=x^4+px^3+2x^2-3x+q` be the given polynomial.
Let `g(x)=(x^2-1)=(x-1) (x+1)`
`(x–1)` and `(x+1)` are factors of `f(x)`
From factor theorem, If `(x–1)` and `(x+1)` are factors of `f(x)` then `f(1)=0` and `f(-1)=0 ` respectively.
`f(1)=0`
`=>(1)^4+p(1)^3+2(1)^2-3(1)+q=0`
`=>1+p+2-3+q=0`
`=>p+q=0` .......(i)
Similarly, `f(-1)=0`
`=>(-1)^4+p(-1)^3+2(-1)^2-3(-1)+q=0`
`=>1-p+2+3+q=0`
`=>q-p+6=0` .......(ii)
Adding equation (i) and equation (ii), we get:
`p+q+q-p+6=0`
`=>2q+6=0`
`=>2q=-6`
`=>q=-3`
Putting value of `q` in equation (i), we get:
`p-3=0`
`=>p=3`
`therefore x^2-1` is divisible by `f(x)` when `p=3` and `q=-3`.