What is the quotien when `(a ^(4) - b ^(4))` is divided by a - b ?

Answer

To find the quotient when \( a^4 - b^4 \) is divided by \( a - b \), we can use the algebraic identity for the difference of squares. Here are the steps to solve the problem:

### Step-by-Step Solution:

1. **Identify the Expression**: We start with the expression \( a^4 - b^4 \).

2. **Use the Difference of Squares Identity**: We know that \( a^4 - b^4 \) can be factored using the difference of squares:
\[ a^4 - b^4 = (a^2 - b^2)(a^2 + b^2) \]

3. **Factor \( a^2 - b^2 \)**: The term \( a^2 - b^2 \) can also be factored further using the difference of squares:
\[ a^2 - b^2 = (a - b)(a + b) \]

4. **Combine the Factors**: Now we can rewrite \( a^4 - b^4 \) as:
\[ a^4 - b^4 = (a - b)(a + b)(a^2 + b^2) \]

5. **Divide by \( a - b \)**: We are interested in the quotient when \( a^4 - b^4 \) is divided by \( a - b \):
\[ \frac{a^4 - b^4}{a - b} = \frac{(a - b)(a + b)(a^2 + b^2)}{a - b} \]

6. **Cancel \( a - b \)**: Since \( a - b \) is in both the numerator and the denominator, we can cancel it out:
\[ = (a + b)(a^2 + b^2) \]

7. **Final Answer**: Therefore, the quotient when \( a^4 - b^4 \) is divided by \( a - b \) is:
\[ (a + b)(a^2 + b^2) \]

### Final Result:
The quotient is \( (a + b)(a^2 + b^2) \).