To analyze and simplify the given algebraic expression [tex]\(27x^3y^2 - 9x^2y^3\)[/tex], let's go through the solution step-by-step.
1. Identify common factors:
- First, look for common numerical factors between the terms. The coefficients [tex]\(27\)[/tex] and [tex]\(9\)[/tex] have a common factor of [tex]\(9\)[/tex].
- Next, identify common variables and their lowest powers. The variable [tex]\(x\)[/tex] appears in both terms, with the lowest power being [tex]\(x^2\)[/tex]. The variable [tex]\(y\)[/tex] also appears in both terms, with the lowest power being [tex]\(y^2\)[/tex].
2. Factor out the common terms:
- The greatest common factor (GCF) of the expression [tex]\(27x^3y^2 - 9x^2y^3\)[/tex] is [tex]\(9x^2y^2\)[/tex].
- Factor [tex]\(9x^2y^2\)[/tex] out of each term:
[tex]\[
27x^3y^2 - 9x^2y^3 = 9x^2y^2 (3x) - 9x^2y^2 (y)
\][/tex]
3. Simplify the expression inside the parentheses:
- Combine the factored terms:
[tex]\[
27x^3y^2 - 9x^2y^3 = 9x^2y^2 (3x - y)
\][/tex]
Thus, the expression [tex]\(27x^3y^2 - 9x^2y^3\)[/tex] can be factored and simplified to:
[tex]\[
9x^2y^2 (3x - y)
\][/tex]
This is the simplified form of the original expression.
