Question

write the partial fraction decomposition for the rational expression. Check your result algebraically by combining the fractions.
1/(x^2 -1)

Answer 1

\[ \large
\frac{1}{x^2-1}
\]First you factor it with difference of squares\[
\frac{1}{(x+1)(x-1)}
\]Now we want this form:\[
\frac{1}{(x+1)(x-1)} = \frac{A}{x+1}+\frac{B}{x-1}
\]We know that\[
A(x-1)+B(x+1)=1
\]Since it was used in order to get the numerator of 1.

Answer 2

We get\[
Ax-A+Bx+B = 1
\]\[
(A+B)x+(B-A) = 1
\]

Answer 3

So \[
(A+B)x+(B-A) = 0x+1
\]Gives us two equations:\[
A+B = 0\quad\quad B-A = 1
\]