Question
Multiply. State any restrictions on the variable.
(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.
(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.
Answers
GPT-4o mini
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) \(x \neq -5, 5, -3, 6\)
### 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) \(x \neq -5, 5, -3, 6\)