Polynomial division for improper rational functions

When a rational Laplace transform $X(s)=N(s)/D(s)$ is improper — numerator degree $\ge$ denominator degree — partial fractions doesn’t apply directly. The first step is to do polynomial long division of $N$ by $D$, splitting $X$ into a polynomial-plus-proper-rational form.

Example

Take $X(s)=s^2/(s+1)$. Numerator degree 2, denominator degree 1. Improper.

Polynomial division: $s^2÷(s+1)=s-1$, remainder $1$. So $s^2=(s-1)(s+1)+1$, and

$X(s)=s-1+\frac{1}{s+1}.$

The proper part $1/(s+1)$ inverse-transforms to $e^{-t}u(t)$.

The polynomial part $s-1$ inverse-transforms to a combination of impulse derivatives and impulses. Since $1\leftrightarrow \delta (t)$ and the time-differentiation property says $sX(s)\leftrightarrow x^{\prime }(t)$, we have $s\leftrightarrow {\delta }^{\prime }(t)$ (an impulse derivative — the limit of an infinitely tall, infinitely narrow doublet).

So $x(t)={\delta }^{\prime }(t)-\delta (t)+e^{-t}u(t)$.

The pattern

If $X(s)$ is improper, the time-domain signal has impulses and impulse derivatives in addition to ordinary functions:

• $1\leftrightarrow \delta (t)$
• $s\leftrightarrow {\delta }^{\prime }(t)$
• $s^2\leftrightarrow {\delta }^{\prime \prime }(t)$
• …

After long division, the polynomial part becomes a sum of such impulse-derivative terms, and the proper-rational part inverse-transforms via partial fractions to ordinary exponentials, polynomials, and damped sinusoids.

When this comes up

For physical signals (continuous functions, possibly with discontinuities but not impulses), the transfer function is always proper — the numerator has fewer derivatives of the input than the denominator has of the output. So this issue almost never arises in physical-system analysis.

It can come up in homework or exam problems where $X(s)$ is given as a formula without an underlying physical context. The check: is $\mathrm{deg}N\ge \mathrm{deg}D$? If yes, long-divide first. If no, partial-fraction directly.

A cleaner restatement

Any rational function in $s$ can be written as

$X(s)=Q(s)+\frac{R(s)}{D(s)},$

with $Q(s)$ a polynomial (the quotient from long division) and $R(s)$ of degree strictly less than $D(s)$ (the remainder). The quotient $Q(s)$ corresponds to a finite combination of $\delta (t)$ and its derivatives; the rational remainder is proper and inverse-transformable by partial fractions.