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.

$\frac{5x^3+2x-1}{x^2-4}$.

Short answer

The rational expression $\frac{5x^3+2x-1}{x^2-4}$is improper.

The sum of a polynomial and a proper rational expression is$5x+\frac{22x-1}{x^2-4}$.

Step-by-step solution

Step 1. Given Information.

The rational expression is,

$\frac{5x^3+2x-1}{x^2-4}$.

Step 2. Improper and Proper Rational Expression.

$\frac{5x^3+2x-1}{x^2-4}$.

The degree of the numerator$=3$.

The degree of the denominator$=2$

Degree of the numerator>Degree of the denominator.

It is an improper rational expression.

Step 3. Sum of Polynomial and Proper Rational Expression.

The improper rational expression is,

$\frac{5x^3+2x-1}{x^2-4}$.

Dividing the expression, we get,

$=5x+\frac{22x-1}{x^2-4}$.

The polynomial is $5x$and the proper rational expression is$\frac{22x-1}{x^2-4}$.