Resolve `(2x+3)/((x-1)(x^(2)+x+1))` into partial fractions.

Answer

To resolve the expression \(\frac{2x + 3}{(x - 1)(x^2 + x + 1)}\) into 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 simpler fractions. Since the denominator consists of a linear factor \((x - 1)\) and a quadratic factor \((x^2 + x + 1)\), we can write:

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

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

### Step 2: Combine the Right Side
Next, we combine the right side into a single fraction:

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

### Step 3: Expand the Numerator
Now, we expand the numerator:

\[ (Ax + B)(x - 1) + C(x^2 + x + 1) = Ax^2 - Ax + Bx - B + Cx^2 + Cx + C \]

Combining like terms, we get:

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

### Step 4: Set the Numerator Equal to the Original Numerator
Now we set the numerator equal to the original numerator \(2x + 3\):

\[ (A + C)x^2 + (-A + B + C)x + (C - B) = 2x + 3 \]

### Step 5: Create a System of Equations
From this equality, we can create a system of equations by comparing coefficients:

1. For \(x^2\): \(A + C = 0\) (1)
2. For \(x\): \(-A + B + C = 2\) (2)
3. For the constant term: \(C - B = 3\) (3)

### Step 6: Solve the System of Equations
From equation (1), we can express \(A\) in terms of \(C\):

\[ A = -C \]

Substituting \(A = -C\) into equations (2) and (3):

From (2):

\[ -(-C) + B + C = 2 \implies C + B + C = 2 \implies 2C + B = 2 \implies B = 2 - 2C \quad (4) \]

From (3):

\[ C - B = 3 \implies C - (2 - 2C) = 3 \implies C - 2 + 2C = 3 \implies 3C - 2 = 3 \implies 3C = 5 \implies C = \frac{5}{3} \]

Now substituting \(C = \frac{5}{3}\) back into equation (1):

\[ A = -C = -\frac{5}{3} = -\frac{5}{3} \]

And substituting \(C\) into equation (4):

\[ B = 2 - 2C = 2 - 2 \cdot \frac{5}{3} = 2 - \frac{10}{3} = \frac{6}{3} - \frac{10}{3} = -\frac{4}{3} \]

### Step 7: Write the Partial Fraction Decomposition
Now we have \(A\), \(B\), and \(C\):

- \(A = -\frac{5}{3}\)
- \(B = -\frac{4}{3}\)
- \(C = \frac{5}{3}\)

Thus, the partial fraction decomposition is:

\[ \frac{2x + 3}{(x - 1)(x^2 + x + 1)} = \frac{-\frac{5}{3}x - \frac{4}{3}}{x^2 + x + 1} + \frac{\frac{5}{3}}{x - 1} \]

### Final Answer
Therefore, the final result is:

\[ \frac{2x + 3}{(x - 1)(x^2 + x + 1)} = \frac{-5x - 4}{3(x^2 + x + 1)} + \frac{5}{3(x - 1)} \]

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