Q. 196 - Intermediate Algebra

Question

In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.
$n^6+512$

Step-by-Step Solution

Answer

The factored form of the given expression is $(n^2+8)(n^4-8n^2+64)$.

Step 1. Given information.

The given expression is:
$n^6+512$

Step 2. Determine the factored form.

The given expression can be written as:
$\begin{array}{rcl}{n}^6+512& =& {\left(n^2\right)}^3+8^3\\ & =& (n^2+8)\left({\left(n^2\right)}^2-\left(n^2\right)\left(8\right)+8^2\right) \left[\because a^3+b^3=(a+b)(a^2-ab+b^2)\right]\\ & =& (n^2+8)\left(n^4-8n^2+64\right)\end{array}$

Step 3. Conclusion.

The factored form of the given expression is $(n^2+8)(n^4-8n^2+64)$.