x³ + y³ over x - y divided by x² - xy + y over x² + xy + y²

The expression you provided can be written as:

\[
\frac{\frac{x^3 + y^3}{x - y}}{\frac{x^2 - xy + y}{x^2 + xy + y^2}}
\]

Let's simplify this step-by-step.

### Step 1: Simplify the numerator.
The numerator is \( \frac{x^3 + y^3}{x - y} \).

Using the identity for the difference of cubes:

\[
x^3 + y^3 = (x + y)(x^2 - xy + y^2)
\]

So,

\[
\frac{x^3 + y^3}{x - y} = \frac{(x + y)(x^2 - xy + y^2)}{x - y}
\]

### Step 2: Simplify the denominator.
The denominator is \( \frac{x^2 - xy + y}{x^2 + xy + y^2} \).

This expression cannot be factored further, so it remains as is.

### Step 3: Combine the expressions.
Now, the entire expression becomes:

\[
\frac{\frac{(x + y)(x^2 - xy + y^2)}{x - y}}{\frac{x^2 - xy + y}{x^2 + xy + y^2}}
\]

This is a division of fractions, which can be written as:

\[
\frac{(x + y)(x^2 - xy + y^2)}{x - y} \times \frac{x^2 + xy + y^2}{x^2 - xy + y}
\]

### Step 4: Cancel common terms.
Notice that \(x^2 - xy + y^2\) appears in both the numerator and the denominator, so we can cancel this term:

\[
\frac{x + y}{x - y} \times \frac{x^2 + xy + y^2}{x^2 - xy + y}
\]

This simplifies to:

\[
\frac{(x + y)(x^2 + xy + y^2)}{(x - y)(x^2 - xy + y)}
\]

### Final Simplified Form:

\[
\frac{(x + y)(x^2 + xy + y^2)}{(x - y)(x^2 - xy + y)}
\]

Would you like further explanation or have any questions?

### Related Questions:
1. What is the geometric significance of \( x^2 + xy + y^2 \) and similar expressions?
2. How does the factorization of sums of cubes work in general?
3. How would this expression behave if \( x = y \)?
4. What happens if \( x \) and \( y \) are complex numbers?
5. Can this type of expression be generalized for higher powers?

### Tip:
Always look for identities such as sum or difference of cubes, which can help simplify complex expressions efficiently.

Explore how to simplify the algebraic expression x³ + y³ over x - y divided by x² - xy + y over x² + xy + y² with a step-by-step breakdown. This problem involves key algebraic concepts like polynomial division, factorization, and the use of the difference and sum of cubes formulas. Ideal for students in Grades 10-12.

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