Multiply. State any restrictions on the variable.

Question

(x^2 -11x+30)/(x^2 -25) x (x^2 +8x+15)/(x^2 -3x-18)
Simplify the rational expression.
=____
What are the restrictions on the variable? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
1)x(=/) ____ (Use a comma to separate answers as needed.)
2)There are no restrictions on x.

Answer 1

To multiply the rational expressions \(\frac{x^2 - 11x + 30}{x^2 - 25} \cdot \frac{x^2 + 8x + 15}{x^2 - 3x - 18}\), we first need to factor each polynomial where possible.

Step 1: Factor the polynomials.

First expression:

1. Factor \(x^2 - 11x + 30\):

   • This factors to \((x - 5)(x - 6)\).

2. Factor \(x^2 - 25\) (difference of squares):

   • This factors to \((x - 5)(x + 5)\).

Second expression:

3. Factor \(x^2 + 8x + 15\):

   • This factors to \((x + 3)(x + 5)\).

4. Factor \(x^2 - 3x - 18\):

   • This factors to \((x - 6)(x + 3)\).

Step 2: Rewrite the multiplication with the simplified factors:

Now we can rewrite the original multiplication:

\[ \frac{(x - 5)(x - 6)}{(x - 5)(x + 5)} \cdot \frac{(x + 3)(x + 5)}{(x - 6)(x + 3)} \]

Step 3: Cancel common factors:

Cancel the common factors in the numerator and denominator:

• \( (x - 5) \) in the numerator and denominator.
• \( (x + 3) \) in the numerator and denominator.
• \( (x - 6) \) in the numerator and denominator.

Resulting expression:

After canceling, the simplified expression is:

\[ \frac{1}{x + 5} \]

Step 4: Identify restrictions on the variable:

To find the restrictions, we must identify values of \(x\) that make the original denominators zero.

1. From \(x^2 - 25 = 0\):

   • \(x - 5 = 0 \Rightarrow x = 5\)
   • \(x + 5 = 0 \Rightarrow x = -5\)

2. From \(x^2 - 3x - 18 = 0\):

   • \(x - 6 = 0 \Rightarrow x = 6\)
   • \(x + 3 = 0 \Rightarrow x = -3\)

Conclusion:

The variable restrictions are \(x \neq -5, 5, -3, 6\).

Thus, the final answer is: \[ \frac{1}{x + 5} \]

Restrictions on \(x\):

• 1. \(x \neq -5, 5, -3, 6\)