Originally Posted by
magentarita Determine whether the given expressions are proper or improper. If improper, rewrite it as the sum of a polynomial and a proper rational expression.
(1) (5x + 2)/(x^3 - 1)
(2) 2x(x^2 + 4)/(x^2 + 1)
Improper Rational Expression
A rational expression in which the degree of the numerator polynomial is greater than or equal to the degree of the denominator polynomial.
Note: Polynomial long division can be used to write an improper rational expression as the sum of a polynomial and a proper rational expression. Synthetic division may also be used for some fractions.
Proper Rational Expression
A rational expression in which the degree of the numerator polynomial is less than the degree of the denominator polynomial.
(1) Proper Rational Expression
(2) Improper Rational Expression
$\displaystyle \frac{2x(x^2+4)}{x^2+1}=\frac{2x^3+8x}{x^2+1}$
Now, perform long division to arrive at:
$\displaystyle 2x+\frac{6x}{x^2+1}$