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8. What is the factored form of [tex]1.728 y^{3} - 125[/tex]?

A. [tex](1.2y - 5)\left(1.44y^{2} + 6y + 25\right)[/tex]

B. [tex](1.2y + 5)\left(1.44y^{2} + 6y - 25\right)[/tex]

C. [tex](1.2y - 5)\left(1.44y^{2} - 6y - 25\right)[/tex]

D. [tex](1.2y + 5)\left(1.44y^{2} - 6y + 25\right)[/tex]


Sagot :

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]
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