Question

find the binomial that completes the factorization. $27y^{3}-512z^{3}=\left(\square\
ight)\left(9y^{2}+24yz+64z^{2}\
ight)$

Explanation:

Step1: Recognize difference of cubes

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

Step2: Identify $a$ and $b$

$27y^3=(3y)^3$, so $a=3y$; $512z^3=(8z)^3$, so $b=8z$

Step3: Match to the given factor

The given second factor is $9y^2+24yz+64z^2=(3y)^2+(3y)(8z)+(8z)^2$, which is $a^2+ab+b^2$. Thus the missing binomial is $a-b$.
<Expression>
$a - b = 3y - 8z$
</Expression>

Answer:

$3y - 8z$