Write the quotient and remainder when we divide :
`(x^(2) - 4x + 4)` by (x - 2)

Answer

To find the quotient and remainder when dividing \( x^2 - 4x + 4 \) by \( x - 2 \), we can use polynomial long division. Let's go through the steps:

### Step 1: Set up the division
We are dividing \( x^2 - 4x + 4 \) (the dividend) by \( x - 2 \) (the divisor).

### Step 2: Divide the first term
Take the first term of the dividend \( x^2 \) and divide it by the first term of the divisor \( x \):
\[ \frac{x^2}{x} = x \]
This is the first term of our quotient.

### Step 3: Multiply and subtract
Now, multiply the entire divisor \( x - 2 \) by the term we just found \( x \):
\[ x \cdot (x - 2) = x^2 - 2x \]
Next, subtract this from the original dividend:
\[ (x^2 - 4x + 4) - (x^2 - 2x) = -4x + 2x + 4 = -2x + 4 \]

### Step 4: Repeat the process
Now, we take the new dividend \( -2x + 4 \) and divide the first term \( -2x \) by the first term of the divisor \( x \):
\[ \frac{-2x}{x} = -2 \]
This is the next term of our quotient.

### Step 5: Multiply and subtract again
Now, multiply the entire divisor \( x - 2 \) by \( -2 \):
\[ -2 \cdot (x - 2) = -2x + 4 \]
Subtract this from the current dividend:
\[ (-2x + 4) - (-2x + 4) = 0 \]

### Step 6: Conclusion
Since the remainder is \( 0 \), we conclude that:
- The quotient is \( x - 2 \)
- The remainder is \( 0 \)

Thus, when dividing \( x^2 - 4x + 4 \) by \( x - 2 \), we have:
- **Quotient:** \( x - 2 \)
- **Remainder:** \( 0 \)

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