The expression (x - y) (x$^2$ + xy + y$^2$) + (x + y) (x$^2$ – xy + y$^2$) – (x + y) (x$^2$ – y$^2$) is equal to

Question

Accepted Answer

To determine why Option 3 is the correct answer for the given expression, let's simplify the expression step by step and verify that it matches the form given in Option 3.

The expression is:

$$
(x - y)(x^2 + xy + y^2) + (x + y)(x^2 - xy + y^2) - (x + y)(x^2 - y^2)
$$

1. **Simplify each part of the expression:**

- **First Term:** $(x - y)(x^2 + xy + y^2)$

Expanding this, we get:
     $$
     = x(x^2 + xy + y^2) - y(x^2 + xy + y^2)
     = x^3 + x^2y + xy^2 - yx^2 - xy^2 - y^3
     = x^3 - y^3 + x^2y - yx^2
     $$

- **Second Term:** $(x + y)(x^2 - xy + y^2)$

Expanding this, we get:
     $$
     = x(x^2 - xy + y^2) + y(x^2 - xy + y^2)
     = x^3 - x^2y + xy^2 + yx^2 - y^2x + y^3
     = x^3 + y^3 - x^2y + yx^2
     $$

- **Third Term:** $(x + y)(x^2 - y^2)$

Expanding this, we get:
     $$
     = x(x^2 - y^2) + y(x^2 - y^2)
     = x^3 - xy^2 + yx^2 - y^3
     $$

2. **Combine all the terms:**

Now, let's combine all the expanded terms:

$$
   (x^3 - y^3 + x^2y - yx^2) + (x^3 + y^3 - x^2y + yx^2) - (x^3 - xy^2 + yx^2 - y^3)
   $$

Simplifying further:
   - Combine like terms:
     $$
     = x^3 - y^3 + x^2y - yx^2 + x^3 + y^3 - x^2y + yx^2 - x^3 + xy^2 - yx^2 + y^3
     $$

- Notice that many terms cancel out:
     $$
     = x^3 - x^3 + y^3 - y^3 + x^2y - x^2y + yx^2 - yx^2 + xy^2 - xy^2 + y^3
     $$

- The remaining terms:
     $$
     = x^3 + y^3 + xy(y - x)
     $$

This matches the form given in Option 3: $x^3 + y^3 + xy(y - x)$.

3. **Why the other options are incorrect:**

- **Option 1:** $x^3 - y^3 + xy(x + y)$
     - This option suggests a different combination of terms that does not match the simplified expression.

- **Option 2:** $y^3 - x^3 + xy(y + x)$
     - This option reverses the signs of $x^3$ and $y^3$ and suggests a different combination of terms.

- **Option 4:** $x^3 + y^3 + xy(x - y)$
     - This option suggests a different combination of terms that does not match the simplified expression.

Therefore, the correct answer is indeed Option 3: $x^3 + y^3 + xy(y - x)$.