The task is to select the correct method for factoring the mathematical expression [tex]\(A^3 - B^3\)[/tex].
Let's analyze each of the given options:
A. Factoring the sum of two cubes
The formula for factoring the sum of two cubes is:
[tex]\[
A^3 + B^3 = (A+B)(A^2 - AB + B^2)
\][/tex]
This doesn't apply to our expression because we have a difference, not a sum.
B. Factoring trinomials by trial and error
This technique is used for expressions of the form [tex]\(ax^2 + bx + c\)[/tex]. Our expression is not a trinomial (it's a binomial), so this option is not correct.
C. Factoring by grouping
Factoring by grouping is useful for polynomials with four terms. Since we have a binomial, this method is not applicable.
D. Factoring out the GCF (Greatest Common Factor)
Factoring out the GCF involves taking a common factor from all terms. Here, there's no common factor between [tex]\(A^3\)[/tex] and [tex]\(B^3\)[/tex] other than 1, which doesn't help in factoring the expression further.
E. Factoring the difference of two cubes
The correct formula for factoring the difference of two cubes is:
[tex]\[
A^3 - B^3 = (A-B)(A^2 + AB + B^2)
\][/tex]
This matches the form of our given expression [tex]\(A^3 - B^3\)[/tex].
F. Factoring the difference of two squares
The formula is:
[tex]\[
A^2 - B^2 = (A+B)(A-B)
\][/tex]
This applies to the difference of squares, but our expression involves cubes, not squares.
G. Factoring perfect square trinomials
These formulas are:
[tex]\[
A^2 + 2AB + B^2 = (A+B)^2 \quad \text{or} \quad A^2 - 2AB + B^2 = (A-B)^2
\][/tex]
Our expression doesn't fit this form, as it involves cubes, not squares.
Thus, the correct method for factoring the expression [tex]\(A^3 - B^3\)[/tex] is:
[tex]\[
\boxed{E. \text{Factoring the difference of two cubes, } A^3 - B^3 = (A - B)(A^2 + AB + B^2)}
\][/tex]
Hence, the correct option is E.
