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

Answer

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

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

### Step 2: Clear the Denominator
To eliminate the denominators, we multiply both sides of the equation by \((x+1)(x+2)\):
\[ x = A(x+2) + B(x+1) \]

### Step 3: Expand the Right Side
Now, we expand the right side:
\[ x = Ax + 2A + Bx + B \]
Combining like terms gives:
\[ x = (A + B)x + (2A + B) \]

### Step 4: Set Up the System of Equations
Now we can equate the coefficients of \(x\) and the constant terms from both sides of the equation:
1. Coefficient of \(x\): \(A + B = 1\)
2. Constant term: \(2A + B = 0\)

### Step 5: Solve the System of Equations
We can solve this system of equations. From the first equation, we can express \(B\) in terms of \(A\):
\[ B = 1 - A \]
Substituting this into the second equation:
\[ 2A + (1 - A) = 0 \]
This simplifies to:
\[ 2A + 1 - A = 0 \implies A + 1 = 0 \implies A = -1 \]
Now substituting \(A = -1\) back into the equation for \(B\):
\[ B = 1 - (-1) = 2 \]

### Step 6: Write the Partial Fraction Decomposition
Now that we have the values for \(A\) and \(B\), we can write the partial fraction decomposition:
\[ \frac{x}{(x+1)(x+2)} = \frac{-1}{x+1} + \frac{2}{x+2} \]

### Final Answer
Thus, the resolved partial fractions are:
\[ \frac{x}{(x+1)(x+2)} = \frac{-1}{x+1} + \frac{2}{x+2} \]

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