\( 24 x^{4} y^{2} z, 18 x^{2} y^{4} \), and \( 12 x^{3} y z \)

Let the three expressions be: \[ a = 24 x^{4} y^{2} z,\quad b = 18 x^{2} y^{4},\quad c = 12 x^{3} y z. \] **Step 1. Factor the numerical coefficients** Break down each numerical coefficient into its prime factors: - \(24 = 2^3 \times 3\) - \(18 = 2 \times 3^2\) - \(12 = 2^2 \times 3\) The greatest common factor (GCF) for the numerical parts is given by taking the smallest power of each prime: - For \(2\): \(\min(3,\,1,\,2)=1\) so \(2^1=2\). - For \(3\): \(\min(1,\,2,\,1)=1\) so \(3^1=3\). Thus, the numerical GCF is: \[ 2 \times 3 = 6. \] **Step 2. Factor the variable \(x\)** Look at the exponents of \(x\) in each term: - \(x^4\) in \(24 x^{4} y^{2} z\) - \(x^2\) in \(18 x^{2} y^{4}\) - \(x^3\) in \(12 x^{3} y z\) The smallest exponent is \(2\), so the \(x\) part of the GCF is: \[ x^2. \] **Step 3. Factor the variable \(y\)** Examine the exponents of \(y\): - \(y^2\) in \(24 x^{4} y^{2} z\) - \(y^4\) in \(18 x^{2} y^{4}\) - \(y^1\) in \(12 x^{3} y z\) (since \(y\) is \(y^1\)) The smallest exponent is \(1\), so the \(y\) part of the GCF is: \[ y. \] **Step 4. Factor the variable \(z\)** Check the exponents of \(z\): - \(z^1\) in \(24 x^{4} y^{2} z\) - \(z^0\) in \(18 x^{2} y^{4}\) (there is no \(z\) factor) - \(z^1\) in \(12 x^{3} y z\) The smallest exponent is \(0\), so \(z\) does not contribute to the GCF. **Step 5. Combine all parts** Combine the numerical and variable factors: \[ \text{GCF} = 6 \times x^2 \times y = 6x^2y. \] Thus, the greatest common factor of the given expressions is: \[ 6x^2y. \]