partial fraction decomposition of 1/x^2-1

The partial fraction decomposition of the rational function \( \frac{1}{x^2 - 1} \) can be done by recognizing that \( x^2 - 1 \) is a difference of squares. Specifically:

\[
x^2 - 1 = (x - 1)(x + 1)
\]

Now, we can express \( \frac{1}{x^2 - 1} \) as a sum of partial fractions:

\[
\frac{1}{x^2 - 1} = \frac{A}{x - 1} + \frac{B}{x + 1}
\]

To find \( A \) and \( B \), we multiply both sides by \( (x - 1)(x + 1) \) to clear the denominators:

\[
1 = A(x + 1) + B(x - 1)
\]

Expanding the right-hand side:

\[
1 = A(x + 1) + B(x - 1) = A x + A + B x - B = (A + B)x + (A - B)
\]

Now, we can equate the coefficients of like terms. From the equation:

\[
(A + B)x + (A - B) = 1
\]

We compare the coefficients of \( x \) and the constant terms:

1. \( A + B = 0 \) (coefficient of \( x \))
2. \( A - B = 1 \) (constant term)

Solving this system of equations:

1. \( A + B = 0 \) implies \( A = -B \)
2. Substituting \( A = -B \) into \( A - B = 1 \):

\[
- B - B = 1 \quad \Rightarrow \quad -2B = 1 \quad \Rightarrow \quad B = -\frac{1}{2}
\]

Now, since \( A = -B \), we have:

\[
A = \frac{1}{2}
\]

Thus, the partial fraction decomposition of \( \frac{1}{x^2 - 1} \) is:

\[
\frac{1}{x^2 - 1} = \frac{1/2}{x - 1} - \frac{1/2}{x + 1}
\]

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### Related Questions:
1. What is the partial fraction decomposition of \( \frac{1}{x^2 - 4} \)?
2. How do you handle partial fraction decomposition when the denominator has repeated roots?
3. Can you decompose \( \frac{2x+3}{x^2-1} \) into partial fractions?
4. What are the steps for decomposing rational functions with quadratic factors in the denominator?
5. How does partial fraction decomposition work for higher-degree polynomials?

### Tip:
When decomposing rational functions, always factor the denominator first. This simplifies the decomposition into simpler fractions.

Learn how to decompose the rational function 1/(x^2 - 1) into partial fractions. This problem involves key algebraic concepts such as the difference of squares and solving linear equations. Suitable for students in Grades 10-12.

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