Divide `(a^(4) - b^(4))` by a - b and find the quotient and remainder .

Answer

To divide \( a^4 - b^4 \) by \( a - b \) and find the quotient and remainder, we can follow these steps:

### Step 1: Recognize the Difference of Squares
The expression \( a^4 - b^4 \) can be recognized as a difference of squares. We can use the identity:
\[ x^2 - y^2 = (x - y)(x + y) \]
In our case, we can express \( a^4 - b^4 \) as:
\[ (a^2)^2 - (b^2)^2 \]
This allows us to apply the difference of squares formula.

### Step 2: Apply the Difference of Squares Formula
Using the formula, we can write:
\[ a^4 - b^4 = (a^2 - b^2)(a^2 + b^2) \]

### Step 3: Factor \( a^2 - b^2 \) Again
Now, we can further factor \( a^2 - b^2 \) using the same difference of squares formula:
\[ a^2 - b^2 = (a - b)(a + b) \]
Thus, we can rewrite \( a^4 - b^4 \) as:
\[ a^4 - b^4 = (a - b)(a + b)(a^2 + b^2) \]

### Step 4: Divide by \( a - b \)
Now, we can divide \( a^4 - b^4 \) by \( a - b \):
\[ \frac{a^4 - b^4}{a - b} = \frac{(a - b)(a + b)(a^2 + b^2)}{a - b} \]
The \( a - b \) terms cancel out:
\[ = (a + b)(a^2 + b^2) \]

### Step 5: Identify the Quotient and Remainder
The quotient from the division is:
\[ \text{Quotient} = (a + b)(a^2 + b^2) \]
Since there are no remaining terms after the division, the remainder is:
\[ \text{Remainder} = 0 \]

### Final Result
Thus, the quotient is \( (a + b)(a^2 + b^2) \) and the remainder is \( 0 \).

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