Factories `125x^(3)+8y^(3)+z^(3)-30xyz.`

Answer

To factor the expression \( 125x^3 + 8y^3 + z^3 - 30xyz \), we can use the identity for the sum of cubes. The identity states that:

\[ x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx) \]

### Step-by-Step Solution:

1. **Identify the terms in the expression**:
The given expression is \( 125x^3 + 8y^3 + z^3 - 30xyz \). We can rewrite \( 125x^3 \) and \( 8y^3 \) in terms of cubes:
- \( 125x^3 = (5x)^3 \)
- \( 8y^3 = (2y)^3 \)

So, we can rewrite the expression as:
\[ (5x)^3 + (2y)^3 + z^3 - 30xyz \]

2. **Recognize the form**:
We see that the expression is in the form of \( a^3 + b^3 + c^3 - 3abc \) where:
- \( a = 5x \)
- \( b = 2y \)
- \( c = z \)

Here, \( 30xyz \) can be rewritten as \( 3(5x)(2y)(z) \).

3. **Apply the identity**:
Using the identity:
\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]
We substitute \( a \), \( b \), and \( c \):
\[ (5x + 2y + z)((5x)^2 + (2y)^2 + z^2 - (5x)(2y) - (5x)(z) - (2y)(z)) \]

4. **Calculate each part**:
- \( (5x)^2 = 25x^2 \)
- \( (2y)^2 = 4y^2 \)
- \( z^2 = z^2 \)
- \( (5x)(2y) = 10xy \)
- \( (5x)(z) = 5xz \)
- \( (2y)(z) = 2yz \)

Putting it all together, we get:
\[ (5x + 2y + z)(25x^2 + 4y^2 + z^2 - 10xy - 5xz - 2yz) \]

5. **Final expression**:
Therefore, the factorization of the original expression is:
\[ (5x + 2y + z)(25x^2 + 4y^2 + z^2 - 10xy - 5xz - 2yz) \]

### Final Answer:
\[ 125x^3 + 8y^3 + z^3 - 30xyz = (5x + 2y + z)(25x^2 + 4y^2 + z^2 - 10xy - 5xz - 2yz) \]