Question

1. for the expression $216y^3 + 0.125$, which factorizations are correct?

a. $(6y + 0.5)(36y^2 - 3y - 0.025)$
b. $(6y - 0.5)(36y^2 + 3y - 0.025)$
c. $(6y - 0.5)(36y^2 - 3y + 0.025)$
d. $(6y + 0.5)(36y^2 - 3y + 0.25)$

Explanation:

Step1: Recognize sum of cubes

$216y^3 + 0.125 = (6y)^3 + (0.5)^3$

Step2: Apply sum of cubes formula

Sum of cubes: $a^3+b^3=(a+b)(a^2-ab+b^2)$
Here $a=6y$, $b=0.5$

Step3: Compute each factor

First factor: $a+b=6y+0.5$
Second factor: $a^2-ab+b^2=(6y)^2-(6y)(0.5)+(0.5)^2=36y^2-3y+0.25$

Step4: Verify match

Result matches option d.

Answer:

d. $(6y + 0.5)(36y^2 - 3y + 0.25)$