Question

Factor:$500p^3+4q^3$.

Short answer

Factors of$500p^3+4q^3$are$4\left(5p+q\right)\left(25p^2+5pq+q^2\right)$.

Step-by-step solution

Step 1. Given information.

Given an expression$500p^3+4q^3$.

Step 2. Factor the common factor.

Common factor of the given expression is 4.

So, the expression can be expressed as $4\left(125p^3+q^3\right)$.

Step 3. Check if the binomial 125p3+q3 is a sum of perfect cubes.

Consider $a=5p$and $b=q$.

It can be seen that $125p^3+q^3={\left(5p\right)}^3+q^3$.

It follows that the expression$125p^3+q^3$is sum of perfect cubes.

Step 4. Use the sum of cubes pattern and simplify.

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

Obtain the result as follows.

$\begin{array}{rcl}4\left(125p^3+q^3\right)& =& 4\left({\left(5p\right)}^3+q^3\right)\\ & =& 4\left(5p+q\right)\left({\left(5p\right)}^2-5pq+q^2\right)\\ & =& 4\left(5p+q\right)\left(25p^2-5pq+q^2\right)\end{array}$

Step 5. Check the result by multiplying the factors.

Verify the result as follows.

$\begin{array}{rcl}4\left(5p+q\right)\left(25p^2-5pq+q^2\right)& =& 4\left(5p\left(25p^2-5pq+q^2\right)+q\left(25p^2-5pq+q^2\right)\right)\\ & =& 4\left(125p^3-25p^{2}q+5p{q}^2+25p^{2}q-5p{q}^2+q^3\right)\\ & =& 4\left(125p^3+q^3\right)\\ & =& 500p^3+4q^3\end{array}$