The sum of the rational expression `(x-3)/(x^(2) +1)` and its reciprocal is

Options

A. `(2x^(2)-6x+10)/(x-3)`
B. `(x^(4) + 3x^(2)-6x+10)/(x^(3) -3x^(2)+x-3)`
C. `(x^(4) +2x^(2) -6x-10)/(x^(3) 3x^(2) +x-3)`
D. None of these

Answer

To find the sum of the rational expression \(\frac{x-3}{x^2 + 1}\) and its reciprocal, we will follow these steps:

### Step 1: Write the given expression and its reciprocal
The given expression is:
\[ \frac{x-3}{x^2 + 1} \]
The reciprocal of this expression is:
\[ \frac{x^2 + 1}{x - 3} \]

### Step 2: Find a common denominator
To add these two fractions, we need a common denominator. The common denominator will be the product of the two denominators:
\[ (x^2 + 1)(x - 3) \]

### Step 3: Rewrite each fraction with the common denominator
Now, we rewrite each fraction with the common denominator:
\[ \frac{x-3}{x^2 + 1} = \frac{(x-3)(x-3)}{(x^2 + 1)(x - 3)} \]
\[ \frac{x^2 + 1}{x - 3} = \frac{(x^2 + 1)(x^2 + 1)}{(x - 3)(x^2 + 1)} \]

### Step 4: Expand the numerators
Now we will expand the numerators:
1. For the first fraction:
\[ (x - 3)(x - 3) = (x - 3)^2 = x^2 - 6x + 9 \]
2. For the second fraction:
\[ (x^2 + 1)(x^2 + 1) = (x^2 + 1)^2 = x^4 + 2x^2 + 1 \]

### Step 5: Combine the numerators
Now we combine the two numerators:
\[ \text{Numerator} = (x^2 - 6x + 9) + (x^4 + 2x^2 + 1) \]
Combining like terms:
\[ = x^4 + (1 + 2)x^2 - 6x + (9 + 1) = x^4 + 3x^2 - 6x + 10 \]

### Step 6: Write the final expression
Now, we can write the final expression for the sum:
\[ \frac{x^4 + 3x^2 - 6x + 10}{(x^2 + 1)(x - 3)} \]

### Final Answer
Thus, the sum of the rational expression \(\frac{x-3}{x^2 + 1}\) and its reciprocal is:
\[ \frac{x^4 + 3x^2 - 6x + 10}{(x^2 + 1)(x - 3)} \]

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