Expression `(x+5)/((x-2)^(2))` has repeated (twice) linear factors in denominator, so find partial fractions.

Answer

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

### Step 1: Identify the form of the partial fractions
Since the denominator \((x - 2)^2\) has a repeated linear factor, we can express the fraction in the form:
\[ \frac{x + 5}{(x - 2)^2} = \frac{A}{x - 2} + \frac{B}{(x - 2)^2} \]
where \(A\) and \(B\) are constants that we need to determine.

### Step 2: Combine the right-hand side
To combine the right-hand side into a single fraction, we find a common denominator:
\[ \frac{A}{x - 2} + \frac{B}{(x - 2)^2} = \frac{A(x - 2) + B}{(x - 2)^2} \]

### Step 3: Set the numerators equal
Now, we set the numerators equal to each other since the denominators are the same:
\[ x + 5 = A(x - 2) + B \]

### Step 4: Expand the right-hand side
Expanding the right-hand side gives:
\[ x + 5 = Ax - 2A + B \]

### Step 5: Rearrange the equation
Rearranging the equation, we can group the terms:
\[ x + 5 = Ax + (B - 2A) \]

### Step 6: Equate coefficients
Now, we equate the coefficients of \(x\) and the constant terms from both sides:
1. For \(x\): \(A = 1\)
2. For the constant term: \(B - 2A = 5\)

### Step 7: Solve for \(B\)
Substituting \(A = 1\) into the second equation:
\[ B - 2(1) = 5 \implies B - 2 = 5 \implies B = 7 \]

### Step 8: Write the partial fractions
Now that we have \(A\) and \(B\), we can write the partial fractions:
\[ \frac{x + 5}{(x - 2)^2} = \frac{1}{x - 2} + \frac{7}{(x - 2)^2} \]

### Final Answer
Thus, the partial fractions of the expression \(\frac{x + 5}{(x - 2)^2}\) are:
\[ \frac{1}{x - 2} + \frac{7}{(x - 2)^2} \]