Factor the polynomial completely. If the polynomial cannot be factored, say it is prime. $$ -x^{2}-18 x-81 $$

Question

UpStudy AI · Accepted Answer

To factor the polynomial $$ -x^{2} - 18x - 81 $$ completely, follow these steps:

1. **Factor Out the Negative Sign:**
   
   Start by factoring out a $$-1$$ from the polynomial:

$$
   -x^{2} - 18x - 81 = -\left( x^{2} + 18x + 81 \right)
   $$

2. **Factor the Quadratic Expression:**
   
   Now, focus on factoring the quadratic expression inside the parentheses:

$$
   x^{2} + 18x + 81
   $$

Look for two numbers that multiply to $$81$$ and add up to $$18$$. The numbers $$9$$ and $$9$$ satisfy this condition because:

$$
   9 \times 9 = 81 \quad \text{and} \quad 9 + 9 = 18
   $$

Therefore, the quadratic can be factored as:

$$
   x^{2} + 18x + 81 = (x + 9)(x + 9) = (x + 9)^2
   $$

3. **Combine the Factors:**
   
   Substitute the factored form back into the expression with the negative sign:

$$
   -\left( x^{2} + 18x + 81 \right) = -\left( (x + 9)^2 \right) = -(x + 9)^2
   $$

**Final Factored Form:**

$$
\boxed{-(x + 9)^{2}}
$$
The polynomial $$ -x^{2} - 18x - 81 $$ factors to $$ -(x + 9)^2 $$.

Factor the polynomial completely. If the polynomial cannot be factored, say it is prime. \( -x^{2}-18 x-81 \)