To solve the problem, let's factor the algebraic expression [tex]\(x^4 y^2 - 6 x^3 y^3\)[/tex].
First, observe the terms in the given expression:
- The first term is [tex]\(x^4 y^2\)[/tex]
- The second term is [tex]\(6 x^3 y^3\)[/tex]
Step 1: Identify the greatest common factor (GCF) of these terms.
- The coefficients are 1 (from [tex]\(x^4 y^2\)[/tex]) and -6 (from [tex]\(6 x^3 y^3\)[/tex]). The GCF of 1 and -6 is 1.
- For the [tex]\(x\)[/tex]-terms: The minimum power of [tex]\(x\)[/tex] in the terms is [tex]\(x^3\)[/tex]; thus [tex]\(x^3\)[/tex] is a common factor.
- For the [tex]\(y\)[/tex]-terms: The minimum power of [tex]\(y\)[/tex] in the terms is [tex]\(y^2\)[/tex]; thus [tex]\(y^2\)[/tex] is a common factor.
So, the GCF of the entire expression [tex]\(x^4 y^2 - 6 x^3 y^3\)[/tex] is [tex]\(x^3 y^2\)[/tex].
Step 2: Factor out the GCF [tex]\(x^3 y^2\)[/tex] from each term.
[tex]\[x^4 y^2 - 6 x^3 y^3 = (x^4 y^2) - (6 x^3 y^3)\][/tex]
Factor out [tex]\(x^3 y^2\)[/tex]:
[tex]\[x^4 y^2 = x^3 y^2 \cdot x\][/tex]
[tex]\[6 x^3 y^3 = x^3 y^2 \cdot 6 y\][/tex]
The expression now becomes:
[tex]\[x^4 y^2 - 6 x^3 y^3 = x^3 y^2 (x) - x^3 y^2 (6 y)\][/tex]
Combine the like terms within the parentheses:
[tex]\[x^3 y^2 (x - 6 y)\][/tex]
However, when we factor [tex]\(x^3 y^2\)[/tex] out again to get the most simplified form possible, we realize we should factor out [tex]\(x^2 y^2\)[/tex] instead because [tex]\(x^4 y^2 - 6 x^3 y^3\)[/tex] can be rethought as:
[tex]\[x^4 y^2 - 6 x^3 y^3 = x^2 y^2(x^2 y - 6 x y^2) = x^2 y^2(x^2 - 6 x y)\][/tex]
Thus, the simplified factored form is:
[tex]\[x^2 y^2 (x^2 - 6 x y)\][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{x^2 y^2 (x^2 - 6 x y)}\][/tex]
