Question

Which of the following is a rational expression whose domain has a restriction of x≠−1?(1 point)%0D%0AResponses%0D%0A%0D%0A6x3+60x%0D%0AStart Fraction 6 x cubed plus 60 over x End Fraction%0D%0A%0D%0Ax√2x+2%0D%0AStart Fraction Start Root x End Root over 2 x plus 2 End Fraction%0D%0A%0D%0Ax+1x3%0D%0AStart Fraction x plus 1 over x cubed End Fraction%0D%0A%0D%0Ax2+6x+56x+6

Answer 1

To find a rational expression with the domain restriction of \( x \neq -1 \), we need to identify which expression has a denominator that becomes zero when \( x = -1 \).

1. **Expression 1:**
\[
\frac{6x^3 + 60}{x}
\]
The denominator is \( x \), which is zero when \( x = 0 \). So, this does not satisfy the restriction \( x \neq -1 \).

2. **Expression 2:**
\[
\frac{\sqrt{x}}{2x + 2}
\]
The denominator is \( 2x + 2 \). Setting this equal to zero:
\[
2x + 2 = 0 \quad \Rightarrow \quad 2x = -2 \quad \Rightarrow \quad x = -1
\]
This expression does have the restriction \( x \neq -1 \).

3. **Expression 3:**
\[
\frac{x + 1}{x^3}
\]
The denominator \( x^3 \) is zero at \( x = 0 \). Thus, it does not satisfy the restriction \( x \neq -1 \).

4. **Expression 4:**
\[
\frac{x^2 + 6x + 5}{6x + 6}
\]
The denominator is \( 6x + 6 \). Setting this equal to zero:
\[
6x + 6 = 0 \quad \Rightarrow \quad 6x = -6 \quad \Rightarrow \quad x = -1
\]
This expression also has the restriction \( x \neq -1 \).

Therefore, both Expression 2 and Expression 4 have the domain restriction \( x \neq -1 \).

However, since the question asks for "a rational expression" rather than multiple expressions, mainly Expression 2 and Expression 4 can be considered valid answers based on the criteria.

To summarize, the rational expressions with a restriction of \( x \neq -1 \) are:

- \(\frac{\sqrt{x}}{2x + 2}\) (Expression 2)
- \(\frac{x^2 + 6x + 5}{6x + 6}\) (Expression 4)

If one answer is required, you would choose either of these two.