Question

Factor the expression completely.

$8x^3-125$

Short answer

The solution$8x^3-125=\left(2x-5\right)\left(4x^2+10x+25\right)$

Step-by-step solution

Step 1. Apply the special product formula of “cube of the difference”.

If A and B are any real numbers or algebraic expressions, then

$\begin{array}{l}{\left(A-B\right)}^3=A^3-3A^{2}B+3A{B}^2-B^3\\ \Rightarrow A^3-B^3=\left(A-B\right)\left(A^2+AB+B^2\right)\end{array}$

Step 2. Analyze the given expression.

Given$8x^3-125$

Here we have:

$A^3={\left(2x\right)}^3,B^3=5^3$

Step 3. Solve the expression.

Substitute the values in the Cube of Difference formula to get:

$\begin{array}{l}8x^3-125={\left(2x\right)}^3-5^3=\left(A-B\right)\left(A^2+AB+B^2\right)\\ \Rightarrow 8x^3-125=\left(2x-5\right)\left({\left(2x\right)}^2+\left(2x\right)\cdot 5+5^2\right)\\ \Rightarrow 8x^3-125=\left(2x-5\right)\left(4x^2+10x+25\right)\end{array}$