`(2x+1)/((x+1)(x-2))` into partial fractions

Answer

To solve the given expression \(\frac{2x + 1}{(x + 1)(x - 2)}\) using partial fractions, we will follow these steps:

### Step 1: Set Up the Partial Fraction Decomposition
We start by expressing the fraction as a sum of partial fractions. Since the denominator is a product of two linear factors, we can write:
\[ \frac{2x + 1}{(x + 1)(x - 2)} = \frac{A}{x + 1} + \frac{B}{x - 2} \]
where \(A\) and \(B\) are constants that we need to determine.

### Step 2: Combine the Right-Hand Side
Next, we will combine the right-hand side over a common denominator:
\[ \frac{A}{x + 1} + \frac{B}{x - 2} = \frac{A(x - 2) + B(x + 1)}{(x + 1)(x - 2)} \]
This gives us:
\[ \frac{2x + 1}{(x + 1)(x - 2)} = \frac{A(x - 2) + B(x + 1)}{(x + 1)(x - 2)} \]

### Step 3: Equate the Numerators
Since the denominators are the same, we can equate the numerators:
\[ 2x + 1 = A(x - 2) + B(x + 1) \]

### Step 4: Expand the Right-Hand Side
Now, we expand the right-hand side:
\[ A(x - 2) + B(x + 1) = Ax - 2A + Bx + B = (A + B)x + (-2A + B) \]
Thus, we have:
\[ 2x + 1 = (A + B)x + (-2A + B) \]

### Step 5: Set Up the System of Equations
Now we can set up a system of equations by comparing coefficients:
1. For the coefficient of \(x\): \(A + B = 2\) (Equation 1)
2. For the constant term: \(-2A + B = 1\) (Equation 2)

### Step 6: Solve the System of Equations
From Equation 1, we can express \(B\) in terms of \(A\):
\[ B = 2 - A \]
Now, substitute \(B\) into Equation 2:
\[ -2A + (2 - A) = 1 \]
Simplifying this gives:
\[ -2A + 2 - A = 1 \implies -3A + 2 = 1 \implies -3A = -1 \implies A = \frac{1}{3} \]

Now substitute \(A\) back into Equation 1 to find \(B\):
\[ \frac{1}{3} + B = 2 \implies B = 2 - \frac{1}{3} = \frac{6}{3} - \frac{1}{3} = \frac{5}{3} \]

### Step 7: Write the Partial Fraction Decomposition
Now that we have \(A\) and \(B\), we can write the partial fraction decomposition:
\[ \frac{2x + 1}{(x + 1)(x - 2)} = \frac{1/3}{x + 1} + \frac{5/3}{x - 2} \]

### Final Answer
Thus, the final answer in terms of partial fractions is:
\[ \frac{1}{3(x + 1)} + \frac{5}{3(x - 2)} \]