Question

Which of the following is a rational expression whose domain has a restriction of  x≠−1 ?(1 point)Responsesx2+6x+56x+6Start Fraction x squared plus 6 x plus 5 over 6 x plus 6 End Fractionx√2x+2Start Fraction Start Root x End Root over 2 x plus 2 End Fractionx+1x3Start Fraction x plus 1 over x cubed End Fraction6x3+60x

Answer 1

To determine which of the given options has a domain restriction of \( x \neq -1 \), we need to look for an expression that has a denominator that becomes zero when \( x = -1 \).

Let's analyze each option:

1. **\( \frac{x^2 + 6x + 5}{6x + 6} \)**:
- Denominator: \( 6x + 6 = 6(x + 1) \)
- This is zero when \( x = -1 \), so there is a restriction \( x \neq -1 \).

2. **\( \frac{\sqrt{x}}{2x + 2} \)**:
- Denominator: \( 2x + 2 = 2(x + 1) \)
- This is also zero when \( x = -1 \), so there is a restriction \( x \neq -1 \).

3. **\( \frac{x + 1}{x^3} \)**:
- Denominator: \( x^3 \)
- This is zero when \( x = 0 \), so there is **no restriction for \( x \neq -1 \)**.

4. **\( 6x^3 + 60x \)**:
- This is an algebraic expression, not a rational expression (no denominator), so there are no restrictions.

Since both the first and second expressions produce the restriction \( x \neq -1 \), the first two options are both valid answers. However, if you are looking for only one option, the first option is typically preferred.

**Therefore, the correct answer is:**

\( \frac{x^2 + 6x + 5}{6x + 6} \) (with the restriction \( x \neq -1 \)).