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

Given:

Given expression is $ax^3 + x^2 - 2x + b$.

$(x + 1)$ and $(x - 1)$ are factors of $ax^3 + x^2 - 2x + b$.

To do:

We have to find the values of $a$ and $b$.

Solution:

We know that,

If $(x-m)$ is a root of $f(x)$ then $f(m)=0$.

Therefore,

$f(-1)=0$

$\Rightarrow a(-1)^3 + (-1)^2 - 2(-1) + b=0$

$\Rightarrow -a+1+2+b=0$

$\Rightarrow a=b+3$...............(i)

$f(1)=0$

$\Rightarrow a(1)^3 + (1)^2 - 2(1) + b=0$

$\Rightarrow a+1-2+b=0$

$\Rightarrow b+3-1+b=0$                   [From (i)]

$\Rightarrow 2b=-2$

$\Rightarrow b=\frac{-2}{2}$

$\Rightarrow b=-1$

$\Rightarrow a=-1+3=2$

The values of $a$ and $b$ are $2$ and $-1$ respectively.