Q. 6.65 - Intermediate Algebra

Question

Factor: $500p^3+4q^3$.

Step-by-Step Solution

Answer

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

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 125 p 3 + q 3 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.

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