Factor the expression: $$ 8y^{3} - 1 $$

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UpStudy AI · Accepted Answer

To factor the expression $$ 8y^{3} - 1 $$, we can recognize it as a **difference of cubes**. Here's how to proceed:

1. **Identify the Cubes:**
   - $$ 8y^3 = (2y)^3 $$
   - $$ 1 = 1^3 $$

2. **Apply the Difference of Cubes Formula:**
   The formula for factoring a difference of cubes is:
   $$
   a^3 - b^3 = (a - b)(a^2 + ab + b^2)
   $$
   Here, $$ a = 2y $$ and $$ b = 1 $$.

3. **Substitute into the Formula:**
   $$
   8y^3 - 1 = (2y)^3 - 1^3 = (2y - 1)\left( (2y)^2 + (2y)(1) + 1^2 \right)
   $$
   Simplifying further:
   $$
   (2y - 1)(4y^2 + 2y + 1)
   $$

4. **Check for Further Factoring:**
   The quadratic $$ 4y^2 + 2y + 1 $$ does not factor neatly over the real numbers because its discriminant is negative:
   $$
   \Delta = (2)^2 - 4(4)(1) = 4 - 16 = -12
   $$
   Since the discriminant is negative, the quadratic cannot be factored further using real numbers.

**Final Factored Form:**
$$
8y^{3} - 1 = (2y - 1)(4y^2 + 2y + 1)
$$

**Answer:**  
After factoring, the expression is (2 y – 1) multiplied by (4y² + 2y + 1). Thus,

(2y−1)(4y²+2y+1)
The expression $$ 8y^{3} - 1 $$ factors to $$ (2y - 1)(4y^2 + 2y + 1) $$.

Factor the expression:

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