To factor the expression [tex]\(1.728 y^3 - 125\)[/tex], we can apply the difference of cubes formula, which is:
[tex]\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\][/tex]
First, we need to identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex] such that [tex]\(1.728 y^3\)[/tex] and [tex]\(125\)[/tex] can be written as perfect cubes.
[tex]\[
1.728 y^3 = (1.2 y)^3 \\
125 = 5^3
\][/tex]
So, the given expression [tex]\(1.728 y^3 - 125\)[/tex] can be written as:
[tex]\((1.2 y)^3 - 5^3\)[/tex]
Applying the difference of cubes formula:
[tex]\[
a = 1.2 y \\
b = 5
\][/tex]
[tex]\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\][/tex]
Now, substituting [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[
1.728 y^3 - 125 = (1.2 y - 5)((1.2 y)^2 + (1.2 y)(5) + 5^2)
\][/tex]
We simplify each term inside the second parenthesis:
[tex]\[
(1.2 y)^2 = 1.44 y^2 \\
(1.2 y)(5) = 6 y \\
5^2 = 25
\][/tex]
Thus:
[tex]\[
1.728 y^3 - 125 = (1.2 y - 5)(1.44 y^2 + 6 y + 25)
\][/tex]
Therefore, the factored form is:
[tex]\[
(1.2 y - 5)(1.44 y^2 + 6 y + 25)
\][/tex]
Among the given choices, this corresponds to the first option:
[tex]\[
(1.2 y - 5)\left(1.44 y^2 + 6 y + 25\right)
\][/tex]
