Resolve into Partial Fractions
(i) `(3x+7)/(x^(2)-3x+2)`

Answer

To resolve the expression \( \frac{3x + 7}{x^2 - 3x + 2} \) into partial fractions, we will follow these steps:

### Step 1: Factor the Denominator
The first step is to factor the denominator \( x^2 - 3x + 2 \).

\[ x^2 - 3x + 2 = (x - 1)(x - 2) \]

### Step 2: Set Up Partial Fractions
Now that we have factored the denominator, we can express the fraction as a sum of partial fractions:

\[ \frac{3x + 7}{(x - 1)(x - 2)} = \frac{A}{x - 1} + \frac{B}{x - 2} \]

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

### Step 3: Clear the Denominator
Multiply both sides by the denominator \( (x - 1)(x - 2) \) to eliminate the fraction:

\[ 3x + 7 = A(x - 2) + B(x - 1) \]

### Step 4: Expand the Right Side
Now, expand the right side:

\[ 3x + 7 = Ax - 2A + Bx - B \]

Combine like terms:

\[ 3x + 7 = (A + B)x + (-2A - B) \]

### Step 5: Set Up a System of Equations
Now we can set up a system of equations by comparing coefficients from both sides:

1. Coefficient of \( x \): \( A + B = 3 \) (Equation 1)
2. Constant term: \( -2A - B = 7 \) (Equation 2)

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

\[ B = 3 - A \]

Substituting \( B \) into Equation 2:

\[ -2A - (3 - A) = 7 \]

Simplifying this:

\[ -2A - 3 + A = 7 \]
\[ -A - 3 = 7 \]
\[ -A = 10 \quad \Rightarrow \quad A = -10 \]

Now substituting \( A \) back into Equation 1 to find \( B \):

\[ -10 + B = 3 \quad \Rightarrow \quad B = 13 \]

### Step 7: Write the Partial Fraction Decomposition
Now that we have \( A \) and \( B \), we can write the partial fraction decomposition:

\[ \frac{3x + 7}{(x - 1)(x - 2)} = \frac{-10}{x - 1} + \frac{13}{x - 2} \]

### Final Answer
Thus, the partial fraction decomposition of \( \frac{3x + 7}{x^2 - 3x + 2} \) is:

\[ \frac{3x + 7}{(x - 1)(x - 2)} = \frac{-10}{x - 1} + \frac{13}{x - 2} \]

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