Q 266 - Intermediate Algebra

Question

In the following exercises, factor completely.   

$64x^3+125y^3$

Step-by-Step Solution

Answer

The expression is factored as $64x^3+125y^3=(4x+5y)(16x^2-20xy+25y^2)$

Step 1. Given Information

Consider the expression $64x^3+125y^3$

The objective is to factor the expression completely.

Step 2. Factor using the sum of cubes

The sum of cubes can be factored as

$a^3+b^3=(a+b)(a^2-ab+b^2)$

For the given binomial $64x^3+125y^3$, the first term $64x^3$ is the cube of $4x$ and the second term $125y^3$ is the cube of $5y$. So it can factored as

$\begin{array}{rcl}64x^3+125y^3& =& {\left(4x\right)}^3+{\left(5y\right)}^3 \\ & =& (4x+5y)\{{\left(4x\right)}^2-(4x\left)\right(5y)+{\left(5y\right)}^2\} \\ & =& (4x+5y)(16x^2-20xy+25y^2)\end{array}$

Step 3. Check the factors

Multiply the factors to check the solution  

$\begin{array}{rcl}(4x+5y)(16x^2-20xy+25y^2)& =& 4x(16x^2-20xy+25y^2)+5y(16x^2-20xy+25y^2)\\ & =& 64x^3-80x^{2}y+100x{y}^2+80x^{2}y-100x{y}^2+125y^3\\ & =& 64x^3+125y^3\end{array}$

And we get the given expression. So the expression is correctly factored.