To find the factors of the expression [tex]\(8x^3 - 27y^3\)[/tex], we start with recognizing that this expression represents a difference of two cubes. The general formula for factoring the difference of cubes is:
[tex]\[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\][/tex]
In our case, we can rewrite [tex]\(8x^3\)[/tex] as [tex]\((2x)^3\)[/tex] and [tex]\(27y^3\)[/tex] as [tex]\((3y)^3\)[/tex]. Therefore, the expression [tex]\(8x^3 - 27y^3\)[/tex] can be expressed as:
[tex]\[
(2x)^3 - (3y)^3
\][/tex]
Applying the difference of cubes formula:
[tex]\[
a = 2x \quad \text{and} \quad b = 3y
\][/tex]
Now we substitute [tex]\(2x\)[/tex] and [tex]\(3y\)[/tex] into the formula:
[tex]\[
(2x)^3 - (3y)^3 = (2x - 3y)((2x)^2 + (2x)(3y) + (3y)^2)
\][/tex]
Simplifying inside the parentheses:
[tex]\[
= (2x - 3y)(4x^2 + 6xy + 9y^2)
\][/tex]
Thus, the expression [tex]\(8x^3 - 27y^3\)[/tex] factors to:
[tex]\[
(2x - 3y)(4x^2 + 6xy + 9y^2)
\][/tex]
Let's match our factors with the given options:
A. [tex]\((2x - 3y)\)[/tex] – Yes, this is a factor.
B. [tex]\((2x + 3y)\)[/tex] – No, this is not a factor.
C. [tex]\((4x^2 + 6xy + 9y^2)\)[/tex] – Yes, this is a factor.
D. [tex]\((4x^2 + 2xy - 6y^2)\)[/tex] – No, this is not a factor.
E. [tex]\((4x^2 - 2xy + 9y^2)\)[/tex] – No, this is not a factor.
So, the correct answers are:
A. [tex]\((2x - 3y)\)[/tex]
C. [tex]\((4x^2 + 6xy + 9y^2)\)[/tex]
