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

Answer

To express the function \(\frac{2x + 1}{(x + 1)(x - 2)}\) in terms of partial fractions, we can follow these steps:

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

### Step 2: Combine the Right Side
To combine the right side into a single fraction, we find a common denominator:
\[ \frac{A}{x + 1} + \frac{B}{x - 2} = \frac{A(x - 2) + B(x + 1)}{(x + 1)(x - 2)} \]
Thus, we have:
\[ \frac{2x + 1}{(x + 1)(x - 2)} = \frac{A(x - 2) + B(x + 1)}{(x + 1)(x - 2)} \]

### Step 3: Set the Numerators Equal
Since the denominators are the same, we can set the numerators equal to each other:
\[ 2x + 1 = A(x - 2) + B(x + 1) \]

### Step 4: Expand the Right Side
Expanding the right side gives:
\[ 2x + 1 = Ax - 2A + Bx + B \]
Combining like terms, we get:
\[ 2x + 1 = (A + B)x + (-2A + B) \]

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

### Step 6: Solve the System of Equations
We have the following system of equations:
1. \(A + B = 2\) (Equation 1)
2. \(-2A + B = 1\) (Equation 2)

From Equation 1, we can express \(B\) in terms of \(A\):
\[ B = 2 - A \]
Substituting this into Equation 2:
\[ -2A + (2 - A) = 1 \]
Simplifying this gives:
\[ -3A + 2 = 1 \implies -3A = -1 \implies A = \frac{1}{3} \]

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

### Final Result
Thus, the partial fraction decomposition is:
\[ \frac{2x + 1}{(x + 1)(x - 2)} = \frac{1/3}{x + 1} + \frac{5/3}{x - 2} \]

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