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.

$\frac{3x^4+x^2-2}{x^3+8}$.

Short answer

The rational expression $\frac{3x^4+x^2-2}{x^3+8}$is improper.

The sum of a polynomial and a proper rational expression is$3x+\frac{x^2-24x-2}{x^3+8}$.

Step-by-step solution

Step 1. Given Information.

The rational expression is,

$\frac{3x^4+x^2-2}{x^3+8}$.

Step 2. Proper or Improper Rational Expression.

$\frac{3x^4+x^2-2}{x^3+8}$.

The degree of the numerator$=4$.

The degree of the denominator$=3$.

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

It is an improper rational expression.

Step 3. Sum of polynomial and proper rational expression.

The improper rational expression is,

$\frac{3x^4+x^2-2}{x^3+8}$.

Add and subtract $24x$in the numerator of the expression,

$$\begin{gathered} =\frac{3x^4+24x-24x+x^2-2}{x^3+8} \\ =\frac{3x^4+24x}{x^3+28}+\frac{x^2-24x-2}{x^3+8} \\ =\frac{3x(x^3+8)}{x^3+8}+\frac{x^2-24x-2}{x^3+8} \\ =3x+\frac{x^2-24x-2}{x^3+8} \end{gathered}$$

The polynomial is$3x$and the proper rational expression is$\frac{x^2-24x-2}{x^3+8}$.