Q. 197 - Intermediate Algebra

Question

In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.
$z^6-27$

Step-by-Step Solution

Answer

The factored form of the given expression is $(z^2-3)(z^4+3z^2+9)$.

Step 1. Given information.

The given expression is:
$z^6-27$

Step 2. Determine the factored form.

The given expression can be written as:

$\begin{array}{rcl}{z}^6-27& =& {\left(z^2\right)}^3-3^3\\ & =& (z^2-3)\left({\left(z^2\right)}^2+\left(z^2\right)\left(3\right)+3^2\right) \left[\because a^3-b^3=(a-b)(a^2+ab+b^2)\right]\\ & =& (z^2-3)\left(z^4+3z^2+9\right)\end{array}$

Step 3. Conclusion.

The factored form of the given expression is $(z^2-3)(z^4+3z^2+9)$.