Question

If x+1 and x-1 are the factors of the polynomial ax^3 + x^2 - 2x +b,find the values of a and b

Answer 1

The polynomial remainder theorem says that dividing a polynomial
`p(x)` by
`x-c` leaves a remainder of
`p(c)=0` if
`x-c` is a factor of
`p(x)`. In this case, check
`c=-1` and
`c=1`.

`a(-1)^3+(-1)^2-2(-1)+b=-a+b+3=0`

`a(1)^3+(1)^2-2(1)+b=a+b-1=0`

From the first equation,

`-a+b+3=0\implies a=b+3`

and substituting into the second gives

`(b+3)+b-1=0\implies2b+2=0\implies b=-1\implies a=2`