Rational functions
A rational function \(R(x)\) is the quotient of two polynomials \(\boldsymbol{P(x)}\) and \(\boldsymbol{Q(x)}\)
\[R(x)=\dfrac{P(x)}{Q(x)};\qquad Q(x)\neq 0\]The simplest rational function is the reciprocal function \(f(x)=\dfrac{1}{x}\). If the degree of the numerator is less than that of the denominator, the rational function is called a ‘proper rational’ function. If not, it is called an ‘improper rational’ function. An improper rational function can always be expressed (with polynomial division) as the sum of a polynomial and a proper rational function.