Question

add these polynomials
(2x/x^2-1)+(3/x-1)

Answer 1

To add the given polynomials \(\frac{2x}{x^2 - 1} + \frac{3}{x - 1}\), we first need to find a common denominator.

### Step 1: Factor the denominators
The first term has a denominator \(x^2 - 1\), which can be factored as follows:

\[
x^2 - 1 = (x - 1)(x + 1)
\]

Thus, the first term becomes:

\[
\frac{2x}{(x - 1)(x + 1)}
\]

The second term has a denominator of \(x - 1\).

### Step 2: Determine the common denominator
The common denominator of \((x - 1)(x + 1)\) and \(x - 1\) is \((x - 1)(x + 1)\).

### Step 3: Adjust the second term
To add the two fractions, we need to express both terms with this common denominator:

The first term is already over the common denominator:

\[
\frac{2x}{(x - 1)(x + 1)}
\]

For the second term, we will multiply the numerator and the denominator by \(x + 1\) to express it over the common denominator:

\[
\frac{3}{x - 1} \cdot \frac{x + 1}{x + 1} = \frac{3(x + 1)}{(x - 1)(x + 1)}
\]

### Step 4: Combine the two fractions
Now we can add the two fractions:

\[
\frac{2x}{(x - 1)(x + 1)} + \frac{3(x + 1)}{(x - 1)(x + 1)} = \frac{2x + 3(x + 1)}{(x - 1)(x + 1)}
\]

### Step 5: Simplify the numerator
Now simplify the numerator:

\[
2x + 3(x + 1) = 2x + 3x + 3 = 5x + 3
\]

### Step 6: Write the final result
Thus, the final result of the addition of the two polynomials is:

\[
\frac{5x + 3}{(x - 1)(x + 1)}
\]

This is the simplified form of the sum of the two polynomials.