Q. 6.63 - Intermediate Algebra

Question

Factor: $8x^3-27y^3$.

Step-by-Step Solution

Answer

Factors of $8x^3-27y^3$ are $\left(2x-3y\right)\left(4x^2-6xy+9y^2\right)$.

Step 1. Given information.

Given an expression $8x^3-27y^3$.

Step 2. Check if the binomial is a difference and write the terms as cubes.

Consider $a=2x$ and $b=3y$.

It can be seen that $8x^3-27y^3={\left(2x\right)}^3-{\left(3y\right)}^3$.

Therefore, $8x^3-27y^3$ is a difference of cubes.

Step 3. Use the difference of cubes pattern and simplify.

Note that $a^3-b^3=\left(a-b\right)\left(a^2+ab+b^2\right)$.

Factor the expression as follows.

$\begin{array}{rcl}8x^3-27y^3& =& {\left(2x\right)}^3-{\left(3y\right)}^3\\ & =& \left(2x-3y\right)\left({\left(2x\right)}^2+\left(2x\right)\left(3y\right)+{\left(3y\right)}^2\right)\\ & =& \left(2x-3y\right)\left(4x^2+6xy+9y^2\right)\end{array}$

Step 4. Check by multiplying the factors.

Verify as follows.

$\begin{array}{rcl}\left(2x-3y\right)\left(4x^2-6xy+9y^2\right)& =& 2x\left(4x^2-6xy+9y^2\right)-3y\left(4x^2-6xy+9y^2\right)\\ & =& 8x^3-12x^{2}y+18x{y}^2-12x^{2}y+18x{y}^2-27y^3\\ & =& 8x^3-27y^3\end{array}$