Resolve the following into partial fractions.
`(x^(2)-3)/((x-2)(x^(2)+1))`

Answer

To resolve the expression \(\frac{x^2 - 3}{(x - 2)(x^2 + 1)}\) into partial fractions, we follow these steps:

### Step 1: Set Up the Partial Fraction Decomposition
We can express the given fraction as:
\[ \frac{x^2 - 3}{(x - 2)(x^2 + 1)} = \frac{A}{x - 2} + \frac{Bx + C}{x^2 + 1} \]
where \(A\), \(B\), and \(C\) are constants that we need to determine.

### Step 2: Clear the Denominator
Multiply both sides by the denominator \((x - 2)(x^2 + 1)\) to eliminate the fraction:
\[ x^2 - 3 = A(x^2 + 1) + (Bx + C)(x - 2) \]

### Step 3: Expand the Right Side
Now, we expand the right side:
\[ x^2 - 3 = A(x^2 + 1) + Bx(x - 2) + C(x - 2) \]
\[ = Ax^2 + A + Bx^2 - 2Bx + Cx - 2C \]
Combine like terms:
\[ = (A + B)x^2 + (-2B + C)x + (A - 2C) \]

### Step 4: Equate Coefficients
Now, we equate the coefficients from both sides of the equation:
1. Coefficient of \(x^2\): \(A + B = 1\)
2. Coefficient of \(x\): \(-2B + C = 0\)
3. Constant term: \(A - 2C = -3\)

### Step 5: Solve the System of Equations
From the first equation, we can express \(A\) in terms of \(B\):
\[ A = 1 - B \]

Substituting \(A\) into the third equation:
\[ (1 - B) - 2C = -3 \]
This simplifies to:
\[ 1 - B - 2C = -3 \implies -B - 2C = -4 \implies B + 2C = 4 \quad \text{(Equation 4)} \]

Now we have two equations:
1. \( -2B + C = 0 \quad \text{(Equation 2)} \)
2. \( B + 2C = 4 \quad \text{(Equation 4)} \)

From Equation 2, we can express \(C\) in terms of \(B\):
\[ C = 2B \]

Substituting \(C\) into Equation 4:
\[ B + 2(2B) = 4 \implies B + 4B = 4 \implies 5B = 4 \implies B = \frac{4}{5} \]

Now substituting back to find \(C\):
\[ C = 2B = 2 \cdot \frac{4}{5} = \frac{8}{5} \]

Now substituting \(B\) back to find \(A\):
\[ A = 1 - B = 1 - \frac{4}{5} = \frac{1}{5} \]

### Step 6: Write the Partial Fraction Decomposition
Now we can write the partial fraction decomposition:
\[ \frac{x^2 - 3}{(x - 2)(x^2 + 1)} = \frac{1/5}{x - 2} + \frac{(4/5)x + (8/5)}{x^2 + 1} \]

### Step 7: Final Expression
Thus, the final expression in partial fractions is:
\[ \frac{1}{5(x - 2)} + \frac{4x + 8}{5(x^2 + 1)} \]