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 difference of cubes

The expression $216y^3 - 0.125$ is a difference of cubes, since $216y^3=(6y)^3$ and $0.125=(0.5)^3$. The formula for difference of cubes is:
$$a^3 - b^3=(a - b)(a^2 + ab + b^2)$$

Step2: Assign values to $a$ and $b$

Let $a=6y$ and $b=0.5$.

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 the factorization

Multiply the factors to confirm:
$$(6y - 0.5)(36y^2 + 3y + 0.25)=216y^3 + 18y^2 + 1.5y - 18y^2 - 1.5y - 0.125=216y^3 - 0.125$$

Answer:

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