Rational Fraction

Rational fractions

If the expressions a and b are polynomials, the algebraic fraction is called a rational algebraic fraction or simply rational fraction. Rational fractions are also known as rational expressions. A rational fraction ${\displaystyle {\tfrac {f(x)}{g(x)}}}$ is called proper if ${\displaystyle \deg f(x)<\deg g(x)}$, and improper otherwise. For example, the rational fraction ${\displaystyle {\tfrac {2x}{x^{2}-1}}}$ is proper, and the rational fractions ${\displaystyle {\tfrac {x^{3}+x^{2}+1}{x^{2}-5x+6}}}$ and ${\displaystyle {\tfrac {x^{2}-x+1}{5x^{2}+3}}}$ are improper. Any improper rational fraction can be expressed as the sum of a polynomial (possibly constant) and a proper rational fraction. In the first example of an improper fraction one has