Question

find the greatest monomial that is a factor of the expression: \\(12f^{4}g^{2} + 15fg^{3} - 3f^{2}g^{2}\\) \\(\circ\\ 3fg^{2}\\) \\(\circ\\ 3g^{2}\\) \\(\circ\\ 6fg^{2}\\) \\(\circ\\ 6f^{4}g^{2}\\)

Explanation:

Step1: Find GCF of coefficients

Coefficients are 12, 15, -3. GCF of 12, 15, 3 is 3.

Step2: Find GCF of variables for f

Exponents of f: 4, 1, 2. Minimum exponent is 1, so \(f^1\).

Step3: Find GCF of variables for g

Exponents of g: 2, 2, 3. Minimum exponent is 2, so \(g^2\).

Step4: Combine GCFs

Multiply coefficient GCF, f GCF, g GCF: \(3\times f\times g^2 = 3fg^2\).

Answer:

\(3fg^2\) (corresponding to the first option: \(3fg^2\))