The rational expression which should be added to `(2x^(3)+1)/(x^(2)+2)` to get `(x^(3)-x^(2)+5)/(x^(2)+2)` is

Options

A. 1
B. `(x^(3)-x^(2)-4)/(x^(2)+2)`
C. `(-x^(3)-x^(2)+4)/(x^(2)+2)`
D. None of these

Answer

To find the rational expression that should be added to \(\frac{2x^3 + 1}{x^2 + 2}\) to obtain \(\frac{x^3 - x^2 + 5}{x^2 + 2}\), we can follow these steps:

### Step 1: Set up the equation
Let the rational expression we need to find be \(a\). We can write the equation as:
\[ a + \frac{2x^3 + 1}{x^2 + 2} = \frac{x^3 - x^2 + 5}{x^2 + 2} \]

### Step 2: Isolate \(a\)
To find \(a\), we can isolate it by subtracting \(\frac{2x^3 + 1}{x^2 + 2}\) from both sides:
\[ a = \frac{x^3 - x^2 + 5}{x^2 + 2} - \frac{2x^3 + 1}{x^2 + 2} \]

### Step 3: Combine the fractions
Since both fractions have the same denominator, we can combine them:
\[ a = \frac{(x^3 - x^2 + 5) - (2x^3 + 1)}{x^2 + 2} \]

### Step 4: Simplify the numerator
Now, we simplify the numerator:
\[ x^3 - x^2 + 5 - 2x^3 - 1 = (x^3 - 2x^3) + (-x^2) + (5 - 1) = -x^3 - x^2 + 4 \]
So we have:
\[ a = \frac{-x^3 - x^2 + 4}{x^2 + 2} \]

### Step 5: Final expression
Thus, the rational expression that should be added is:
\[ \frac{-x^3 - x^2 + 4}{x^2 + 2} \]