Identify the least common denominator of $$ \frac{x+3}{x^{2}+11 x+18} $$ and $$ \frac{-2 x}{x^{2}+6 x-27} $$

Question

UpStudy AI · Accepted Answer

To find the **least common denominator (LCD)** of the two rational expressions:

$$
\frac{x + 3}{x^{2} + 11x + 18} \quad \text{and} \quad \frac{-2x}{x^{2} + 6x - 27},
$$

follow these steps:

### 1. Factor the Denominators

**First Denominator:** $$ x^{2} + 11x + 18 $$

- Look for two numbers that multiply to **18** and add up to **11**.
- These numbers are **9** and **2**.
  
$$
x^{2} + 11x + 18 = (x + 9)(x + 2)
$$

**Second Denominator:** $$ x^{2} + 6x - 27 $$

- Look for two numbers that multiply to **-27** and add up to **6**.
- These numbers are **9** and **-3**.
  
$$
x^{2} + 6x - 27 = (x + 9)(x - 3)
$$

### 2. Identify the Least Common Denominator (LCD)

The denominators factored are:

$$
(x + 9)(x + 2) \quad \text{and} \quad (x + 9)(x - 3)
$$

The LCD is the product of all **distinct** factors, taking the highest power of each:

$$
\text{LCD} = (x + 9)(x + 2)(x - 3)
$$

### Final Answer

$$
\boxed{\, (x + 9)(x + 2)(x - 3)\,}
$$
The least common denominator is $$ (x + 9)(x + 2)(x - 3) $$.

Identify the least common denominator of \( \frac{x+3}{x^{2}+11 x+18} \) and \( \frac{-2 x}{x^{2}+6 x-27} \)