Question
Real Tutor Solution
Tutor-Verified Answer
Answer
\( 8x^{3} - 1 = (2x - 1)(4x^{2} + 2x + 1) \)
Solution
To factor the expression \( 8x^{3} - 1 \), observe that it is a **difference of cubes**.
Recall the formula for factoring a difference of cubes:
\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]
Here, \( 8x^{3} \) can be written as \( (2x)^3 \) and \( 1 \) as \( 1^3 \). Applying the formula:
\[
8x^{3} - 1 = (2x)^3 - 1^3 = (2x - 1)\left((2x)^2 + (2x)(1) + 1^2\right)
\]
Simplifying the terms inside the parentheses:
\[
(2x - 1)(4x^2 + 2x + 1)
\]
The quadratic \( 4x^2 + 2x + 1 \) does not factor further over the real numbers.
**Final Factored Form:**
\[
8x^{3} - 1 = (2x - 1)(4x^2 + 2x + 1)
\]
Answer: \((2x - 1)\big(4\,x^{2} + 2\,x + 1\,\big)\)
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
To factor \( 8x^3 - 1 \), we can recognize it as a difference of cubes. It can be rewritten as \( (2x)^3 - 1^3 \). Using the difference of cubes formula \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \), where \( a = 2x \) and \( b = 1 \), we can factor as follows: \[ 8x^3 - 1 = (2x - 1)((2x)^2 + (2x)(1) + (1)^2) = (2x - 1)(4x^2 + 2x + 1) \] And there you have it: the fully factored form of \( 8x^3 - 1 \) is \( (2x - 1)(4x^2 + 2x + 1) \)!
