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

Answer

To resolve the expression \( \frac{9}{(x-1)(x+2)^2} \) into partial fractions, we will follow these steps:

### Step 1: Set up the partial fraction decomposition
We start by expressing the fraction in terms of its partial fractions. Since we have a linear factor \( (x-1) \) and a repeated linear factor \( (x+2)^2 \), we can write:

\[ \frac{9}{(x-1)(x+2)^2} = \frac{A}{x-1} + \frac{B}{x+2} + \frac{C}{(x+2)^2} \]

where \( A \), \( B \), and \( C \) are constants that we need to determine.

### Step 2: Clear the denominators
Next, we multiply both sides by the common denominator \( (x-1)(x+2)^2 \) to eliminate the fractions:

\[ 9 = A(x+2)^2 + B(x-1)(x+2) + C(x-1) \]

### Step 3: Expand the right-hand side
Now we expand the right-hand side:

1. Expand \( A(x+2)^2 \):
\[ A(x^2 + 4x + 4) = Ax^2 + 4Ax + 4A \]

2. Expand \( B(x-1)(x+2) \):
\[ B(x^2 + x - 2) = Bx^2 + Bx - 2B \]

3. Expand \( C(x-1) \):
\[ Cx - C \]

Combining these, we have:
\[ 9 = (A + B)x^2 + (4A + B + C)x + (4A - 2B - C) \]

### Step 4: Set up the system of equations
Now we equate the coefficients from both sides of the equation. Since the left-hand side has no \( x^2 \) or \( x \) terms, we can set up the following equations:

1. Coefficient of \( x^2 \): \( A + B = 0 \)
2. Coefficient of \( x \): \( 4A + B + C = 0 \)
3. Constant term: \( 4A - 2B - C = 9 \)

### Step 5: Solve the system of equations
From the first equation \( A + B = 0 \), we can express \( B \) in terms of \( A \):
\[ B = -A \]

Substituting \( B = -A \) into the second equation:
\[ 4A - A + C = 0 \implies 3A + C = 0 \implies C = -3A \]

Now substituting \( B = -A \) and \( C = -3A \) into the third equation:
\[ 4A - 2(-A) - (-3A) = 9 \implies 4A + 2A + 3A = 9 \implies 9A = 9 \implies A = 1 \]

Now substituting \( A = 1 \) back to find \( B \) and \( C \):
\[ B = -1 \quad \text{and} \quad C = -3 \]

### Step 6: Write the partial fraction decomposition
Now that we have the values of \( A \), \( B \), and \( C \), we can write the partial fraction decomposition:

\[ \frac{9}{(x-1)(x+2)^2} = \frac{1}{x-1} - \frac{1}{x+2} - \frac{3}{(x+2)^2} \]

### Final Answer
Thus, the partial fraction decomposition of \( \frac{9}{(x-1)(x+2)^2} \) is:

\[ \frac{9}{(x-1)(x+2)^2} = \frac{1}{x-1} - \frac{1}{x+2} - \frac{3}{(x+2)^2} \]