Question

find the binomial that completes the factorization.
$r^3 + 512 = (\square)(r^2 - 8r + 64)$

Explanation:

Step1: Recall sum of cubes formula

The sum of cubes formula is $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.

Step2: Identify $a$ and $b$

Here, $a^3 = r^3$ so $a = r$; $b^3 = 512$ so $b = 8$ (since $8^3 = 512$).

Step3: Find the binomial factor

The binomial factor is $a + b = r + 8$.

Step4: Verify the quadratic term

Check: $(r + 8)(r^2 - 8r + 64) = r^3 - 8r^2 + 64r + 8r^2 - 64r + 512 = r^3 + 512$, which matches the left side.

Answer:

$r + 8$