`(x^(2) - 4x + 4) // (x - 2) = ?`

Options

A. (x - 2)
B. (x + 2)
C. (2 - x)
D. `(2 + x + x^(2))`

Answer

To solve the division of the algebraic expression \((x^{2} - 4x + 4)\) by \((x - 2)\), we can follow these steps:

### Step 1: Set up the division
We want to divide \(x^{2} - 4x + 4\) by \(x - 2\).

### Step 2: Divide the leading terms
The leading term of the dividend \(x^{2}\) is divided by the leading term of the divisor \(x\):
\[ \frac{x^{2}}{x} = x \]
This means we will multiply the entire divisor \(x - 2\) by \(x\).

### Step 3: Multiply and subtract
Now, multiply \(x\) by the entire divisor:
\[ x(x - 2) = x^{2} - 2x \]
Next, subtract this result from the original polynomial:
\[ (x^{2} - 4x + 4) - (x^{2} - 2x) = -4x + 2x + 4 = -2x + 4 \]

### Step 4: Repeat the process
Now, we need to divide \(-2x\) by \(x\):
\[ \frac{-2x}{x} = -2 \]
Now, multiply the entire divisor \(x - 2\) by \(-2\):
\[ -2(x - 2) = -2x + 4 \]
Subtract this from \(-2x + 4\):
\[ (-2x + 4) - (-2x + 4) = 0 \]

### Step 5: Conclusion
Since the remainder is \(0\), we conclude that:
\[ \frac{x^{2} - 4x + 4}{x - 2} = x - 2 \]

### Final Answer:
\[ \frac{x^{2} - 4x + 4}{x - 2} = x - 2 \]

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