Resolve into Partial Fractions
(iv) `(x^(4))/((x-1)(x-2))`

Answer

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

### Step 1: Check the degree of the numerator and denominator
The degree of the numerator \( x^4 \) is 4, and the degree of the denominator \( (x-1)(x-2) \) is 2. Since the degree of the numerator is greater than the degree of the denominator, we need to perform polynomial long division first.

### Step 2: Perform polynomial long division
We divide \( x^4 \) by \( (x-1)(x-2) = x^2 - 3x + 2 \).

1. Divide the leading term: \( \frac{x^4}{x^2} = x^2 \).
2. Multiply \( x^2 \) by \( (x^2 - 3x + 2) \):
\[ x^2(x^2 - 3x + 2) = x^4 - 3x^3 + 2x^2 \]
3. Subtract this from \( x^4 \):
\[ x^4 - (x^4 - 3x^3 + 2x^2) = 3x^3 - 2x^2 \]

4. Now, divide \( 3x^3 \) by \( x^2 \):
\[ \frac{3x^3}{x^2} = 3x \]
5. Multiply \( 3x \) by \( (x^2 - 3x + 2) \):
\[ 3x(x^2 - 3x + 2) = 3x^3 - 9x^2 + 6x \]
6. Subtract:
\[ 3x^3 - 2x^2 - (3x^3 - 9x^2 + 6x) = 7x^2 - 6x \]

7. Now, divide \( 7x^2 \) by \( x^2 \):
\[ \frac{7x^2}{x^2} = 7 \]
8. Multiply \( 7 \) by \( (x^2 - 3x + 2) \):
\[ 7(x^2 - 3x + 2) = 7x^2 - 21x + 14 \]
9. Subtract:
\[ 7x^2 - 6x - (7x^2 - 21x + 14) = 15x - 14 \]

Thus, the result of the division is:
\[ x^2 + 3x + 7 + \frac{15x - 14}{(x-1)(x-2)} \]

### Step 3: Set up the partial fraction decomposition
Now we need to decompose \( \frac{15x - 14}{(x-1)(x-2)} \):
\[ \frac{15x - 14}{(x-1)(x-2)} = \frac{A}{x-1} + \frac{B}{x-2} \]

### Step 4: Clear the denominators
Multiply through by \( (x-1)(x-2) \):
\[ 15x - 14 = A(x-2) + B(x-1) \]

### Step 5: Expand and collect like terms
Expanding the right side:
\[ 15x - 14 = Ax - 2A + Bx - B = (A + B)x + (-2A - B) \]

### Step 6: Set up equations from coefficients
From the coefficients, we have:
1. \( A + B = 15 \)
2. \( -2A - B = -14 \)

### Step 7: Solve the equations
From equation (1):
\[ B = 15 - A \]
Substituting into equation (2):
\[ -2A - (15 - A) = -14 \]
\[ -2A - 15 + A = -14 \]
\[ -A - 15 = -14 \implies -A = 1 \implies A = -1 \]
Substituting \( A = -1 \) back into equation (1):
\[ -1 + B = 15 \implies B = 16 \]

### Step 8: Write the final partial fraction decomposition
Thus, we have:
\[ \frac{x^4}{(x-1)(x-2)} = x^2 + 3x + 7 - \frac{1}{x-1} + \frac{16}{x-2} \]

### Final Answer:
\[ \frac{x^4}{(x-1)(x-2)} = x^2 + 3x + 7 - \frac{1}{x-1} + \frac{16}{x-2} \]