if `(x+1)` and `(x-1)` are factors of `p(x)=ax^3+x^2-2x+b` find the values of `a & b`

Explanation

1409720 Let `f(x)=ax^3+x^2−2x+b`
`g(x)=x+1`
`q(x)=x−1`
If `g(x)` is a factor of `f(x)`, zero of `g(x)` will satisfy `f(x)`
`=>−x+1=0`
`=>x=−1`
`f(−1)=0`
`=>a(−1)^3+(−1)^2−2(−1)+b=0`
`−a+1+2+b=0`
`−a+3+b=0`
`=>a−b=3` .......(1)
Also if `q(x)` is a factor of `f(x)`, zero of q(x) will satisfy `f(x)`
`=>−x−1=0`
`=>x=1`
`f(1)=0`
`a(1)^3+(1)^2−2(1)+b=0`
`=>a+1−2+b=0`
`=>a+b−1=0`
`=>a+b=1` ......(2)
Adding equations (1) and (2)
`a−b+a+b=3+1`
`=>2a=4`
`=>a=2`
`=>a+b=1`
`⇒b=−1`
`therefore a=2` and `b=−1`