Resolve into Partial Fractions
(i) `(42-19x)/((x^2+1)(x-4))`

Answer

To resolve the expression \( \frac{42 - 19x}{(x^2 + 1)(x - 4)} \) into partial fractions, we will follow these steps:

### Step 1: Set up the Partial Fraction Decomposition
We start by expressing the given fraction as a sum of partial fractions. Since the denominator consists of a quadratic term \( (x^2 + 1) \) and a linear term \( (x - 4) \), we can express it as:

\[ \frac{42 - 19x}{(x^2 + 1)(x - 4)} = \frac{Ax + B}{x^2 + 1} + \frac{C}{x - 4} \]

### Step 2: Clear the Denominator
Next, we multiply both sides by the denominator \( (x^2 + 1)(x - 4) \) to eliminate the fractions:

\[ 42 - 19x = (Ax + B)(x - 4) + C(x^2 + 1) \]

### Step 3: Expand the Right Side
Now we expand the right-hand side:

\[ (Ax + B)(x - 4) = Ax^2 - 4Ax + Bx - 4B \]
\[ C(x^2 + 1) = Cx^2 + C \]

Combining these, we get:

\[ 42 - 19x = (A + C)x^2 + (-4A + B)x + (-4B + C) \]

### Step 4: Equate Coefficients
Now we equate the coefficients of both sides:

1. Coefficient of \( x^2 \): \( A + C = 0 \) (1)
2. Coefficient of \( x \): \( -4A + B = -19 \) (2)
3. Constant term: \( -4B + C = 42 \) (3)

### Step 5: Solve the System of Equations
From equation (1), we can express \( C \) in terms of \( A \):

\[ C = -A \]

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

Substituting into (2):

\[ -4A + B = -19 \quad \text{(2)} \]

Substituting into (3):

\[ -4B - A = 42 \quad \text{(3)} \]

Now we have two equations:

1. \( -4A + B = -19 \) (2)
2. \( -4B - A = 42 \) (3)

From equation (2), we can express \( B \):

\[ B = -19 + 4A \]

Substituting \( B \) into equation (3):

\[ -4(-19 + 4A) - A = 42 \]
\[ 76 - 16A - A = 42 \]
\[ 76 - 17A = 42 \]
\[ -17A = 42 - 76 \]
\[ -17A = -34 \]
\[ A = 2 \]

Now substituting \( A = 2 \) back to find \( C \):

\[ C = -A = -2 \]

Now substituting \( A = 2 \) into the equation for \( B \):

\[ B = -19 + 4(2) = -19 + 8 = -11 \]

### Step 6: Write the Partial Fractions
Now we have \( A = 2 \), \( B = -11 \), and \( C = -2 \). Thus, we can write the partial fractions as:

\[ \frac{42 - 19x}{(x^2 + 1)(x - 4)} = \frac{2x - 11}{x^2 + 1} - \frac{2}{x - 4} \]

### Final Answer
The resolved partial fractions are:

\[ \frac{42 - 19x}{(x^2 + 1)(x - 4)} = \frac{2x - 11}{x^2 + 1} - \frac{2}{x - 4} \]

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