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

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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)
After factoring out the GCF, the polynomial becomes: 9(3x⁷ + 9x⁶ − 5x² + 2)

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

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