## Expanding and Simplifying (x + y + z)³ - x³ - y³ - z³

The expression (x + y + z)³ - x³ - y³ - z³ is a fascinating algebraic expression that can be expanded and simplified to reveal interesting patterns and relationships. Let's delve into the process of doing so:

### Expanding the Cube

First, we expand the cube of the trinomial (x + y + z):

(x + y + z)³ = (x + y + z)(x + y + z)(x + y + z)

This multiplication can be done in a systematic way, but it involves a significant amount of terms. The result is:

(x + y + z)³ = x³ + y³ + z³ + 3x²y + 3x²z + 3xy² + 3xz² + 3y²z + 3yz² + 6xyz

### Substitution and Simplification

Now, we substitute this expanded form back into the original expression:

(x + y + z)³ - x³ - y³ - z³ = (x³ + y³ + z³ + 3x²y + 3x²z + 3xy² + 3xz² + 3y²z + 3yz² + 6xyz) - x³ - y³ - z³

Notice that the x³, y³, and z³ terms cancel out. This leaves us with:

3x²y + 3x²z + 3xy² + 3xz² + 3y²z + 3yz² + 6xyz

### Factoring and Final Result

We can factor out a 3 from each term:

3(x²y + x²z + xy² + xz² + y²z + yz² + 2xyz)

This expression can be further simplified by grouping terms:

3[(x²y + xy²) + (x²z + xz²) + (y²z + yz²) + 2xyz]

Finally, factoring out common factors in each group, we arrive at:

**3[xy(x + y) + xz(x + z) + yz(y + z) + 2xyz]**

This is the simplified form of the original expression.

### Significance and Applications

This expression has significance in various fields, including:

**Algebraic identities:**This simplified form highlights the relationship between the cube of a trinomial and its individual terms.**Geometric interpretations:**The expression can be related to volumes of geometric figures, particularly in situations involving three-dimensional shapes.**Advanced mathematics:**The manipulation of similar expressions plays a crucial role in fields like abstract algebra and algebraic geometry.

The process of expanding, simplifying, and factoring this expression demonstrates the power of algebraic techniques for revealing hidden patterns and relationships in mathematical expressions.