Factor the following: $$ 8 x^{3}-1 $$

Question

UpStudy AI · Accepted Answer

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: $$(2x - 1)\big(4\,x^{2} + 2\,x + 1\,\big)$$
$$ 8x^{3} - 1 = (2x - 1)(4x^{2} + 2x + 1) $$

Factor the following: \( 8 x^{3}-1 \)