Resolve the following into partial fractions.
`(x^(2))/ ((x-1)(x-2))`

Answer

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

### Step 1: Identify the form of the partial fractions
Since the degree of the numerator (2) is equal to the degree of the denominator (2), we need to express the fraction as a polynomial plus a proper fraction. We can write:
\[ \frac{x^2}{(x-1)(x-2)} = A + \frac{B}{x-1} + \frac{C}{x-2} \]
where \( A \) is a constant, and \( B \) and \( C \) are the coefficients of the fractions.

### Step 2: Multiply through by the denominator
To eliminate the denominator, we multiply both sides by \( (x-1)(x-2) \):
\[ x^2 = A(x-1)(x-2) + B(x-2) + C(x-1) \]

### Step 3: Expand the right-hand side
Expanding the right-hand side:
\[ x^2 = A(x^2 - 3x + 2) + B(x - 2) + C(x - 1) \]
This simplifies to:
\[ x^2 = Ax^2 - 3Ax + 2A + Bx - 2B + Cx - C \]
Combining like terms gives:
\[ x^2 = Ax^2 + (-3A + B + C)x + (2A - 2B - C) \]

### Step 4: Set up equations for coefficients
Now, we can equate the coefficients from both sides:
1. For \( x^2 \): \( A = 1 \)
2. For \( x \): \( -3A + B + C = 0 \)
3. For the constant term: \( 2A - 2B - C = 0 \)

### Step 5: Substitute \( A \) into the equations
From the first equation, we have \( A = 1 \). Substituting \( A \) into the other equations:
1. \( -3(1) + B + C = 0 \) simplifies to \( B + C = 3 \) (Equation 1).
2. \( 2(1) - 2B - C = 0 \) simplifies to \( 2 - 2B - C = 0 \) or \( C = 2 - 2B \) (Equation 2).

### Step 6: Solve the system of equations
Substituting Equation 2 into Equation 1:
\[ B + (2 - 2B) = 3 \]
This simplifies to:
\[ 2 - B = 3 \implies B = -1 \]
Now substituting \( B = -1 \) back into Equation 2:
\[ C = 2 - 2(-1) = 2 + 2 = 4 \]

### Step 7: Write the final partial fraction decomposition
Now we have \( A = 1 \), \( B = -1 \), and \( C = 4 \). Therefore, we can write:
\[ \frac{x^2}{(x-1)(x-2)} = 1 - \frac{1}{x-1} + \frac{4}{x-2} \]

### Final Answer
Thus, the partial fraction decomposition of \( \frac{x^2}{(x-1)(x-2)} \) is:
\[ 1 - \frac{1}{x-1} + \frac{4}{x-2} \]

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