To determine the greatest common factor (GCF) of the polynomial [tex]\(-18 x^4 y - 9 x y^3 + 3 x y^2\)[/tex], we will follow these steps:
1. Identify the coefficients:
- The coefficients in the given polynomial are [tex]\(-18\)[/tex], [tex]\(-9\)[/tex], and [tex]\(3\)[/tex].
2. Find the GCF of the coefficients:
- The GCF of [tex]\(-18\)[/tex], [tex]\(-9\)[/tex], and [tex]\(3\)[/tex] is [tex]\(3\)[/tex].
3. Determine the variables and their exponents:
- The terms in the polynomial are [tex]\(-18 x^4 y\)[/tex], [tex]\(-9 x y^3\)[/tex], and [tex]\(3 x y^2\)[/tex].
- For the variable [tex]\(x\)[/tex]:
- The exponents are [tex]\(4\)[/tex], [tex]\(1\)[/tex], and [tex]\(1\)[/tex].
- For the variable [tex]\(y\)[/tex]:
- The exponents are [tex]\(1\)[/tex], [tex]\(3\)[/tex], and [tex]\(2\)[/tex].
4. Find the lowest power of each variable:
- For [tex]\(x\)[/tex], the lowest exponent among [tex]\(4\)[/tex], [tex]\(1\)[/tex], and [tex]\(1\)[/tex] is [tex]\(1\)[/tex].
- For [tex]\(y\)[/tex], the lowest exponent among [tex]\(1\)[/tex], [tex]\(3\)[/tex], and [tex]\(2\)[/tex] is [tex]\(1\)[/tex].
5. Combine the GCF of the coefficients and the lowest powers of the variables:
- The GCF of the coefficients is [tex]\(3\)[/tex].
- The lowest power of [tex]\(x\)[/tex] is [tex]\(x^1\)[/tex] or simply [tex]\(x\)[/tex].
- The lowest power of [tex]\(y\)[/tex] is [tex]\(y^1\)[/tex] or simply [tex]\(y\)[/tex].
Hence, the greatest common factor of the polynomial [tex]\(-18 x^4 y - 9 x y^3 + 3 x y^2\)[/tex] is:
[tex]\[ 3xy \][/tex]
