`8x^(3) +1` =______

Options

A. `(2x+1)(4x^(2)-2x+1)`
B. `(2x-1)(4x^(2)-2x+1)`
C. `(2x-1)(4x^(2)+2x+1)`
D. `(2x+1)(4x^(2)+2x+1)`

Answer

To factorize the expression \(8x^3 + 1\), we can recognize that it is a sum of cubes. We can rewrite \(8x^3\) as \((2x)^3\) and \(1\) as \(1^3\).

### Step-by-Step Solution:

1. **Identify the expression as a sum of cubes**:
\[ 8x^3 + 1 = (2x)^3 + 1^3 \]

2. **Use the formula for the sum of cubes**:
The formula for the sum of cubes is:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
Here, \(a = 2x\) and \(b = 1\).

3. **Apply the formula**:
Substitute \(a\) and \(b\) into the formula:
\[ (2x + 1)((2x)^2 - (2x)(1) + (1)^2) \]

4. **Calculate each part**:
- Calculate \((2x)^2\):
\[ (2x)^2 = 4x^2 \]
- Calculate \(-(2x)(1)\):
\[ -(2x)(1) = -2x \]
- Calculate \((1)^2\):
\[ (1)^2 = 1 \]

5. **Combine the results**:
Now, substitute these values back into the expression:
\[ (2x + 1)(4x^2 - 2x + 1) \]

6. **Final Factorized Form**:
Therefore, the factorized form of \(8x^3 + 1\) is:
\[ 8x^3 + 1 = (2x + 1)(4x^2 - 2x + 1) \]

### Final Answer:
\[ 8x^3 + 1 = (2x + 1)(4x^2 - 2x + 1) \]

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