Question

Simplify the rational expression. State any restrictions on the variable. x2+3x−28/x2−16 (1 point) Responses x+7x+4; x≠4, x≠−4 x+7x+4; x≠4, x≠−4 x+7x−4; x≠4, x≠−4 x+7x−4; x≠4, x≠−4 x+7x+4; x≠4 x+7x+4; x≠4 x+7x+4; x≠−4, x≠−7

Answer 1

To simplify the rational expression \(\frac{x^2 + 3x - 28}{x^2 - 16}\), we start by factoring both the numerator and the denominator.

1. **Factor the numerator** \(x^2 + 3x - 28\):
- We look for two numbers that multiply to \(-28\) and add to \(3\).
- The numbers \(7\) and \(-4\) fit this requirement.
- Therefore, we can factor the numerator as \((x + 7)(x - 4)\).

2. **Factor the denominator** \(x^2 - 16\):
- This is a difference of squares, which factors as \((x - 4)(x + 4)\).

So we can rewrite the expression:

\[
\frac{x^2 + 3x - 28}{x^2 - 16} = \frac{(x + 7)(x - 4)}{(x - 4)(x + 4)}
\]

3. **Cancel common factors**:
- We can cancel \((x - 4)\) from the numerator and denominator (as long as \(x \neq 4\)):

\[
\frac{(x + 7)\cancel{(x - 4)}}{\cancel{(x - 4)}(x + 4)} = \frac{x + 7}{x + 4}
\]

4. **Identify restrictions on the variable**:
- The original denominator \(x^2 - 16\) cannot be zero. Hence, we set \(x^2 - 16 = 0\):
- \(x - 4 = 0 \implies x = 4\)
- \(x + 4 = 0 \implies x = -4\)
- Therefore, the restrictions on the variable are \(x \neq 4\) and \(x \neq -4\).

Putting it all together, the simplified expression is:

\[
\frac{x + 7}{x + 4}
\]

And the restrictions are:

\(x \neq 4, x \neq -4\).

So, the correct answer from the options provided is:

**x + 7 / x + 4; x ≠ 4, x ≠ -4**.