Factorize following experessios
(i) `8x^(3)-27y^(3)" (ii)"8x^3-125y^3+2x-5y`

Answer

Let's solve the given expressions step by step.

### Part (i): Factorize \( 8x^3 - 27y^3 \)

1. **Identify the cubes**:
\[ 8x^3 = (2x)^3 \quad \text{and} \quad 27y^3 = (3y)^3 \]
Therefore, we can rewrite the expression as:
\[ 8x^3 - 27y^3 = (2x)^3 - (3y)^3 \]

2. **Apply the difference of cubes formula**:
The formula for the difference of cubes is:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
Here, \( a = 2x \) and \( b = 3y \).

3. **Substitute \( a \) and \( b \) into the formula**:
\[ (2x - 3y)((2x)^2 + (2x)(3y) + (3y)^2) \]

4. **Calculate each term**:
- \( (2x)^2 = 4x^2 \)
- \( (2x)(3y) = 6xy \)
- \( (3y)^2 = 9y^2 \)

5. **Combine the terms**:
\[ 8x^3 - 27y^3 = (2x - 3y)(4x^2 + 6xy + 9y^2) \]

### Final Result for Part (i):
\[ 8x^3 - 27y^3 = (2x - 3y)(4x^2 + 6xy + 9y^2) \]

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### Part (ii): Factorize \( 8x^3 - 125y^3 + 2x - 5y \)

1. **Identify the cubes**:
\[ 8x^3 = (2x)^3 \quad \text{and} \quad 125y^3 = (5y)^3 \]
We can rewrite the expression as:
\[ 8x^3 - 125y^3 + 2x - 5y = (2x)^3 - (5y)^3 + 2x - 5y \]

2. **Apply the difference of cubes formula**:
Using the difference of cubes formula:
\[ (2x - 5y)((2x)^2 + (2x)(5y) + (5y)^2) \]

3. **Calculate each term**:
- \( (2x)^2 = 4x^2 \)
- \( (2x)(5y) = 10xy \)
- \( (5y)^2 = 25y^2 \)

4. **Combine the terms**:
\[ (2x - 5y)(4x^2 + 10xy + 25y^2) \]

5. **Factor out the common term \( (2x - 5y) \)**:
Now, we also have \( 2x - 5y \) in the expression:
\[ (2x - 5y)(4x^2 + 10xy + 25y^2 + 1) \]

### Final Result for Part (ii):
\[ 8x^3 - 125y^3 + 2x - 5y = (2x - 5y)(4x^2 + 10xy + 25y^2 + 1) \]

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