Q. 202 - Intermediate Algebra

Question

In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.
$27x^3-64y^3$

Step-by-Step Solution

Answer

The factored form of the given expression is $(3x-4y)(9x^2+12xy+16y^2)$.

Step 1. Given information.

The given expression is:
$27x^3-64y^3$

Step 2. Determine the factored form.

The given expression can be written as:
$\begin{array}{rcl}27x^3-64y^3& =& {\left(3x\right)}^3-{\left(4y\right)}^3 \\ & =& (3x-4y)({\left(3x\right)}^2+(3x\left)\right(4y)+{\left(4y\right)}^2) \left[\because a^3-b^3=(a-b)(a^2+ab+b^2)\right]\\ & =& (3x-4y)(9x^2+12xy+16y^2)\end{array}$

Step 3. Conclusion.

The factored form of the given expression is $(3x-4y)(9x^2+12xy+16y^2)$.