Answer
$A = \frac{1}{2}; B = -\frac{1}{2}$
Work Step by Step
The given equation is:
$$\dfrac{1}{(x-1)(x+1)} =\dfrac{A}{(x-1)}+\dfrac{B}{(x+1)} ~~~(1)$$
On multiplying equation (1) with $(x-1) (x+1)$, we obtain:
$$1 =A(x+1)+B(x-1)~~~(2)$$
Now, we will put $x=-1$ in order to eliminate $A$ from equation (2), we obtain:
$$1 =A(-1+1)+B(-1-1) \\ B = -\dfrac{1}{2}$$
Next, we will put $x=1$ in order to eliminate $B$ from equation (2), we obtain:
$$1 =A(1+1)+B(1-1) \\ A = \dfrac{1}{2}$$
Our required results are:
$$A = \dfrac{1}{2}; B = -\dfrac{1}{2}$$
