Questions: Factor (8 n^3-p^3) completely.

Transcript text: Factor $8 n^{3}-p^{3}$ completely.

Solution

Solution Steps

To factor the expression $8n^3 - p^3$, we recognize it as a difference of cubes. The formula for factoring a difference of cubes $a^3 - b^3$ is given by: \[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\]

In this case, $8n^3$ can be written as $(2n)^3$ and $p^3$ remains $p^3$. Therefore, we can apply the difference of cubes formula with $a = 2n$ and $b = p$.

Solution Approach

Identify $a$ and $b$ such that $a^3 = 8n^3$ and $b^3 = p^3$.
Apply the difference of cubes formula: $(a - b)(a^2 + ab + b^2)$.

Step 1: Identify the Expression

We start with the expression $8n^3 - p^3$. This expression is recognized as a difference of cubes, where $a = 2n$ and $b = p$.

Step 2: Apply the Difference of Cubes Formula

Using the difference of cubes formula: \[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \] we substitute $a$ and $b$: \[ 8n^3 - p^3 = (2n - p)((2n)^2 + (2n)(p) + p^2) \]

Step 3: Simplify the Factors

Calculating the components:

$a - b = 2n - p$
$a^2 = (2n)^2 = 4n^2$
$ab = (2n)(p) = 2np$
$b^2 = p^2$

Thus, we can express the second factor as: \[ 4n^2 + 2np + p^2 \]

Final Answer

Combining these results, we have the complete factorization: \[ 8n^3 - p^3 = -( -2n + p)(4n^2 + 2np + p^2) \] Therefore, the final answer is: \[ \boxed{8n^3 - p^3 = -( -2n + p)(4n^2 + 2np + p^2)} \]

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