Question

select all correct factorizations for (0.216y^{3}+\frac{1}{27}):
a. ((0.6y - \frac{1}{3})(0.36y^{2}+0.2y - \frac{1}{9}))
b. ((0.6y - \frac{1}{3})(0.36y^{2}-0.2y - \frac{1}{9}))
c. ((0.6y + \frac{1}{3})(0.36y^{2}-0.2y - \frac{1}{9}))
d. ((0.6y + \frac{1}{3})(0.36y^{2}-0.2y + \frac{1}{9}))

identify all correct factorizations for (512y^{3}+27):
a. ((8y - 3)(64y^{2}-24y + 9))
b. ((8y - 3)(64y^{2}+24y - 9))
c. ((8y + 3)(64y^{2}+24y + 9))
d. ((8y + 3)(64y^{2}-24y + 9))

Explanation:

First Problem: Factorize $0.216y^3 + \frac{1}{27}$

Step1: Recognize sum of cubes

$0.216y^3 = (0.6y)^3$, $\frac{1}{27}=(\frac{1}{3})^3$

Step2: Apply sum of cubes formula

Sum of cubes: $a^3+b^3=(a+b)(a^2-ab+b^2)$

Step3: Substitute $a=0.6y, b=\frac{1}{3}$

$(0.6y)^3+(\frac{1}{3})^3=(0.6y+\frac{1}{3})[(0.6y)^2-(0.6y)(\frac{1}{3})+(\frac{1}{3})^2]$

Step4: Simplify the quadratic term

$(0.6y)^2=0.36y^2$, $(0.6y)(\frac{1}{3})=0.2y$, $(\frac{1}{3})^2=\frac{1}{9}$
Result: $(0.6y + \frac{1}{3})(0.36y^2 - 0.2y + \frac{1}{9})$

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Second Problem: Factorize $512y^3 + 27$

Step1: Recognize sum of cubes

$512y^3=(8y)^3$, $27=3^3$

Step2: Apply sum of cubes formula

Sum of cubes: $a^3+b^3=(a+b)(a^2-ab+b^2)$

Step3: Substitute $a=8y, b=3$

$(8y)^3+3^3=(8y+3)[(8y)^2-(8y)(3)+3^2]$

Step4: Simplify the quadratic term

$(8y)^2=64y^2$, $(8y)(3)=24y$, $3^2=9$
Result: $(8y + 3)(64y^2 - 24y + 9)$

Answer:

d. $(0.6y + \frac{1}{3})(0.36y^2 - 0.2y + \frac{1}{9})$