Resolve `(x^(2)-3x+5)/((x-2)^(3))` into partial fractions

Answer

To resolve the expression \(\frac{x^2 - 3x + 5}{(x - 2)^3}\) into partial fractions, we will follow these steps:

### Step 1: Set Up the Partial Fraction Decomposition
We start by expressing the given fraction in terms of its partial fractions:
\[ \frac{x^2 - 3x + 5}{(x - 2)^3} = \frac{A}{x - 2} + \frac{B}{(x - 2)^2} + \frac{C}{(x - 2)^3} \]
where \(A\), \(B\), and \(C\) are constants we need to determine.

### Step 2: Clear the Denominator
Multiply both sides by \((x - 2)^3\) to eliminate the denominator:
\[ x^2 - 3x + 5 = A(x - 2)^2 + B(x - 2) + C \]

### Step 3: Expand the Right Side
Now, we expand the right side:
\[ A(x - 2)^2 = A(x^2 - 4x + 4) = Ax^2 - 4Ax + 4A \]
\[ B(x - 2) = Bx - 2B \]
Combining these, we have:
\[ x^2 - 3x + 5 = Ax^2 + (-4A + B)x + (4A - 2B + C) \]

### Step 4: Compare Coefficients
Now we will compare coefficients from both sides of the equation:
- Coefficient of \(x^2\): \(A = 1\)
- Coefficient of \(x\): \(-4A + B = -3\)
- Constant term: \(4A - 2B + C = 5\)

### Step 5: Solve for Constants
1. From \(A = 1\), substitute into the second equation:
\[ -4(1) + B = -3 \implies B = -3 + 4 = 1 \]

2. Substitute \(A = 1\) and \(B = 1\) into the third equation:
\[ 4(1) - 2(1) + C = 5 \implies 4 - 2 + C = 5 \implies C = 5 - 2 = 3 \]

### Step 6: Write the Partial Fraction Decomposition
Now that we have \(A\), \(B\), and \(C\):
- \(A = 1\)
- \(B = 1\)
- \(C = 3\)

We can write the partial fraction decomposition:
\[ \frac{x^2 - 3x + 5}{(x - 2)^3} = \frac{1}{x - 2} + \frac{1}{(x - 2)^2} + \frac{3}{(x - 2)^3} \]

### Final Answer
Thus, the resolved expression in partial fractions is:
\[ \frac{1}{x - 2} + \frac{1}{(x - 2)^2} + \frac{3}{(x - 2)^3} \]