Question

Add the following rational expressions: 2x/x2+4x−5+1/x+5 . Enter response with terms arranged in order from largest degree to smallest degree.

Answer 1

To add the rational expressions \(\frac{2x}{x^2 + 4x - 5}\) and \(\frac{1}{x + 5}\), we first need to find a common denominator.

1. **Factor the denominator of the first term:**
\[
x^2 + 4x - 5 = (x + 5)(x - 1)
\]
So we rewrite the first term:
\[
\frac{2x}{(x + 5)(x - 1)}
\]

2. **Identify the common denominator:**
The common denominator between \((x + 5)(x - 1)\) and \((x + 5)\) is \((x + 5)(x - 1)\).

3. **Rewriting the second term with the common denominator:**
\[
\frac{1}{x + 5} = \frac{1 \cdot (x - 1)}{(x + 5)(x - 1)} = \frac{x - 1}{(x + 5)(x - 1)}
\]

4. **Combine the two fractions:**
Now we can add the two fractions:
\[
\frac{2x}{(x + 5)(x - 1)} + \frac{x - 1}{(x + 5)(x - 1)} = \frac{2x + (x - 1)}{(x + 5)(x - 1)}
\]

5. **Simplify the numerator:**
Combine like terms in the numerator:
\[
2x + (x - 1) = 2x + x - 1 = 3x - 1
\]

6. **Write the final expression:**
Therefore, the sum of the rational expressions is:
\[
\frac{3x - 1}{(x + 5)(x - 1)}
\]

So the final answer, with terms arranged from largest degree to smallest degree, is:
\[
\frac{3x - 1}{(x + 5)(x - 1)}
\]