The remainder when `x^4 - y^4` is divided by x - y is:

Options

A. 0
B. `x + y`
C. `x^2 - y^2`
D. `2y^4`

Answer

To find the remainder when \( x^4 - y^4 \) is divided by \( x - y \), we can follow these steps:

### Step 1: Factor the expression \( x^4 - y^4 \)
We can recognize that \( x^4 - y^4 \) is a difference of squares. It can be factored as follows:
\[ x^4 - y^4 = (x^2 - y^2)(x^2 + y^2) \]

### Step 2: Further factor \( x^2 - y^2 \)
The term \( x^2 - y^2 \) is also a difference of squares and can be factored further:
\[ x^2 - y^2 = (x - y)(x + y) \]
Thus, we can rewrite \( x^4 - y^4 \) as:
\[ x^4 - y^4 = (x - y)(x + y)(x^2 + y^2) \]

### Step 3: Divide by \( x - y \)
Now, we need to divide \( x^4 - y^4 \) by \( x - y \):
\[ \frac{x^4 - y^4}{x - y} = \frac{(x - y)(x + y)(x^2 + y^2)}{x - y} \]
Since \( x - y \) is a common factor in the numerator and the denominator, we can cancel it out:
\[ = (x + y)(x^2 + y^2) \]

### Step 4: Determine the remainder
When we divide \( x^4 - y^4 \) by \( x - y \), we see that the expression is completely divisible by \( x - y \). Therefore, the remainder is:
\[ \text{Remainder} = 0 \]

### Final Answer
The remainder when \( x^4 - y^4 \) is divided by \( x - y \) is \( \boxed{0} \).