Factor out the GCF from the polynomial. \[ 27 x^{7}+81 x^{6}-45 x^{2}+18 \]

To factor out the **Greatest Common Factor (GCF)** from the polynomial: \[ 27x^{7} + 81x^{6} - 45x^{2} + 18 \] **Step 1: Determine the GCF of the coefficients** - **27** can be factored as \( 3^3 \) - **81** can be factored as \( 3^4 \) - **45** can be factored as \( 3^2 \times 5 \) - **18** can be factored as \( 2 \times 3^2 \) The GCF of the coefficients is the highest power of 3 common to all, which is \( 3^2 = 9 \). **Step 2: Determine the GCF of the variables** - The terms \( 27x^{7} \) and \( 81x^{6} \) have \( x \) with exponents 7 and 6 respectively. - The terms \( -45x^{2} \) and \( 18 \) either have \( x^2 \) or no \( x \) at all. Since the term \( 18 \) does not contain \( x \), the GCF for the variables is \( x^0 = 1 \) (meaning no variable factor). **Step 3: Factor out the GCF** Factor out **9** from each term: \[ \begin{align*} 27x^{7} + 81x^{6} - 45x^{2} + 18 &= 9 \left( \frac{27x^{7}}{9} + \frac{81x^{6}}{9} - \frac{45x^{2}}{9} + \frac{18}{9} \right) \\ &= 9 \left( 3x^{7} + 9x^{6} - 5x^{2} + 2 \right) \end{align*} \] **Final Factored Form:** \[ 9(3x^{7} + 9x^{6} - 5x^{2} + 2) \] Answer: After factoring, the polynomial is 9 multiplied by (3 x⁷ + 9x⁶ – 5x² + 2). Thus, 9(3x⁷ + 9x⁶ − 5x² + 2)