Q. 12 - Precalculus Enhanced with Graphing Utilities

Question

Tell whether the given rational expression is proper or improper. If improper, rewrite it as the sum of a polynomial and a proper rational expression.

$\frac{2x(x^2+4)}{x^2+1}$.

Step-by-Step Solution

Answer

The rational expression $\frac{2x(x^2+4)}{x^2+1}$is improper.

The sum of a polynomial and a proper rational expression is  $2x+\frac{\mathit{6}x}{x^2+1}$.

Step 1. Given Information.

The rational expression is,

$\frac{2x(x^2+4)}{x^2+1}$.

Step 2. Simplifying the rational expression.

The rational expression is,

$\frac{2x(x^2+4)}{x^2+1}$

Simplifying the expression,

$=\frac{2x^3+8x}{x^2+1}$.

Step 3. Improper or Proper Rational Expression.

$\frac{2x^3+8x}{x^2+1}$

The degree of the numerator$=3$.

The degree of the denominator$=2$.

The degree of the numerator > The degree of the denominator.

It is an improper rational expression.

Step 4. Decomposition of Rational Expression.

The improper rational expression is,

$\frac{2x^3+8x}{x^2+1}$.

Splitting the term and solving,

$$\begin{gathered} =\frac{2x(x^2+4)}{x^2+1} \\ =\frac{2x(x^2+1+3)}{x^2+1} \\ =\frac{2x(x^2+1)}{x^2+1}+\frac{6x}{x^2+1} \\ =2x+\frac{6x}{x^2+1} \end{gathered}$$