Chapter 11.5, Problem 12AYU

To determine

To express: The expression $\frac{2x\left(x^2+4\right)}{x^2+1}$ as the sum of a polynomial and a proper rational expression if it is an improper fraction.

Answer to Problem 12AYU

The expression $\frac{2x\left(x^2+4\right)}{x^2+1}$ is an improper rational expression. It is expressed as $\frac{2x\left(x^2+4\right)}{x^2+1}=2x+\frac{6x}{x^2+1}$ .

Explanation of Solution

Given information:

The expression $\frac{2x\left(x^2+4\right)}{x^2+1}$ .

Formula used:

Improper rational expression of the form $\frac{P}{Q}$ where Q has non repeated linear factors of the form $\left(x-a_1\right)\left(x-a_2\right)\cdots \left(x-a_n\right)$ then $\frac{P}{Q}$ is expressed as $\frac{P\left(x\right)}{Q\left(x\right)}=\frac{A_1}{x-a_1}+\frac{A_2}{x-a_2}+\cdots +\frac{A_n}{x-a_n}$ .

Calculation:

Consider the expression $\frac{2x\left(x^2+4\right)}{x^2+1}$ . Rewrite the expression as $\frac{2x^3+8x}{x^2+1}$

Observe that the degree of the numerator of the rational expression is greater than the denominator so the provided expression is improper.

When a polynomial is divided by its factor then dividend is the product of divisor and quotient increased by remainder.

Apply the method of long division.

  $\begin{array}{l}{x}^2+1\begin{array}{c}\hfill 2x\\ \hfill \overline{)2x^3+8x}\end{array}\\ \begin{array}{ccc}& & 2x^3+2x\end{array}\\ \begin{array}{ccc}& & \begin{array}{cc}& 6x\end{array}\end{array}\end{array}$

Now, since $\text{Dividend}=\text{Quotient}\times \text{Divisor}+\text{Remainder}$ .

Therefore,

  $\frac{2x\left(x^2+4\right)}{x^2+1}=2x+\frac{6x}{x^2+1}$

It cannot be further decomposed as $x^2+1$ cannot be factored further.

Thus, the expression $\frac{2x\left(x^2+4\right)}{x^2+1}$ is an improper rational expression. It is expressed as $\frac{2x\left(x^2+4\right)}{x^2+1}=2x+\frac{6x}{x^2+1}$ .