Let's factor the expression [tex]\(3x^4y^3 - 15x^2y^2 + 6xy\)[/tex] completely, step-by-step.
1. Identify common factors: First, we look for the greatest common factor (GCF) of all the terms in the expression.
- The terms are [tex]\(3x^4y^3\)[/tex], [tex]\(-15x^2y^2\)[/tex], and [tex]\(6xy\)[/tex].
- The numerical coefficients are 3, -15, and 6. The GCF of these coefficients is 3.
- For the variables:
- The lowest power of [tex]\(x\)[/tex] in all terms is [tex]\(x\)[/tex].
- The lowest power of [tex]\(y\)[/tex] in all terms is [tex]\(y\)[/tex].
2. Factor out the GCF: We factor out [tex]\(3xy\)[/tex] from each term:
[tex]\[
3x^4y^3 - 15x^2y^2 + 6xy = 3xy(x^3y^2 - 5xy + 2)
\][/tex]
3. Check the remaining expression for further factorization: The expression inside the parentheses is [tex]\( x^3 y^2 - 5xy + 2 \)[/tex].
- Since [tex]\(x^3 y^2 - 5xy + 2\)[/tex] no longer has any common factors or clear ways to factor further using integers and simple expressions, it appears this polynomial is already simplified as far as it can go without more complex factorization techniques.
Thus, the completely factored form of the given expression is:
[tex]\[
3xy (x^3 y^2 - 5xy + 2)
\][/tex]
