State the given function.
This computer science problem involves algorithmic thinking and programming concepts. The solution below explains the approach, logic, and implementation step by step.
ScanSolve AI Answer
Step 1: State the given function.
The function to be expressed in partial fractions is:
$$ f(x) = \frac{2}{(x-1)(x+1)} $$
Step 2: Set up the partial fraction decomposition.
We assume the form:
$$ \frac{2}{(x-1)(x+1)} = \frac{A}{x-1} + \frac{B}{x+1} $$
To find $A$ and $B$, multiply both sides by $(x-1)(x+1)$:
$$ 2 = A(x+1) + B(x-1) $$
Step 3: Solve for the constants $A$ and $B$.
To find $A$, set $x=1$:
$$ 2 = A(1+1) + B(1-1) $$
$$ 2 = 2A + 0 $$
$$ 2 = 2A $$
$$ A = 1 $$
To find $B$, set $x=-1$:
$$ 2 = A(-1+1) + B(-1-1) $$
$$ 2 = 0 + B(-2) $$
$$ 2 = -2B $$
$$ B = -1 $$
Step 4: Write the final partial fraction expression.
Substitute the values of $A$ and $B$ back into the decomposition:
$$ f(x) = \frac{1}{x-1} + \frac{-1}{x+1} $$
$$ f(x) = \frac{1}{x-1} - \frac{1}{x+1} $$
The partial fraction decomposition of $f(x)$ is $\boxed{\frac{1}{x-1} - \frac{1}{x+1}}$.
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