Q. 203 - Intermediate Algebra

Question

In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.
$216a^3+125b^3$

Step-by-Step Solution

Answer

The factored form of the given expression is $(6a+5b)(36a^2-30ab+25b^2)$.

Step 1. Given information.

The given expression is:
$216a^3+125b^3$

Step 2. Determine the factored form.

The given expression can be written as:
$\begin{array}{rcl}216a^3+125b^3& =& {\left(6a\right)}^3+{\left(5b\right)}^3\\ & =& (6a+5b)({\left(6a\right)}^2-(6a\left)\right(5b)+{\left(5b\right)}^2) \left[\because a^3+b^3=(a+b)(a^2-ab+b^2)\right]\\ & =& (6a+5b)(36a^2-30ab+25b^2)\end{array}$

Step 3. Conclusion.

The factored form of the given expression is $(6a+5b)(36a^2-30ab+25b^2)$.