Question

Which rational expression should be added to $\frac{x^3-1}{x^2+2}$ to get $\frac{3x^2+2x^2+4}{x^2+2}$?

• A. $\frac{2x^3+2x^2+5}{x^2+2}$
• B. $\frac{2x^3-2x^2+5}{x^2+2}$
• C. $\frac{2x^3+2x^2+5}{x^2-2}$
• D. None of these

Answer 1

The correct option is A $\frac{2x^3+2x^2+5}{x^2+2}$

Let the rational expression be $A$ that must be added to $\frac{x^3-1}{x^2+2}$ to get $\frac{3x^3+2x^2+4}{x^2+2}$, therefore, we have:

$\frac{x^3-1}{x^2+2}+A=\frac{3x^3+2x^2+4}{x^2+2}$

$\Rightarrow A=\frac{3x^3+2x^2+4}{x^2+2}-\frac{x^3-1}{x^2+2}$

$\Rightarrow A=\frac{(3x^3+2x^2+4)-(x^3-1)}{x^2+2}$

$\Rightarrow A=\frac{3x^3+2x^2+4-x^3+1}{x^2+2}$

$\Rightarrow A=\frac{2x^3+2x^2+5}{x^2+2}$

Hence, $\frac{2x^3+2x^2+5}{x^2+2}$ should be added to $\frac{x^3-1}{x^2+2}$ to get $\frac{3x^3+2x^2+4}{x^2+2}$.