Question

find the binomial that completes the factorization.
$q^3 - 1000 = (\quad)(q^2 + 10q + 100)$

Explanation:

Step1: Recall the difference of cubes formula

The formula for the difference of cubes is $x^3 - y^3=(x - y)(x^2+xy + y^2)$.

Step2: Identify \(x\) and \(y\) in the given expression

In the expression $q^3 - 1000$, we can see that $x = q$ and $y^3=1000$, so $y = \sqrt[3]{1000}=10$.

Step3: Apply the difference of cubes formula

Using the formula $x^3 - y^3=(x - y)(x^2+xy + y^2)$ with $x = q$ and $y = 10$, we get $q^3-1000=(q - 10)(q^2 + 10q+100)$.

Answer:

\(q - 10\)