Recognizing Improper Rational Functions

Why Degree Comparison Comes First

Before applying partial fractions to a rational function, first check whether the numerator has degree less than the denominator. This quick comparison tells you whether the fraction is already in the form needed for a proper rational part, or whether polynomial division is needed first.

The comparison is about the degrees of the whole numerator and denominator polynomials. It does not depend on whether those polynomials are expanded or factored. Equal degrees count as improper, not proper. Common factors may change how the fraction looks, but they do not change the degree comparison that determines its classification.

Degrees and the Properness Test

The degree of a nonzero polynomial is the greatest exponent of its variable with a nonzero coefficient. For example, \(4x^3-2x+7\) has degree \(3\), while the nonzero constant polynomial \(5\) has degree \(0\). The zero polynomial is assigned degree \(-\infty\) for the purpose of the comparison below.

A rational function is written as \(P(x)/Q(x)\), where \(P\) and \(Q\) are polynomials and \(Q\) is not the zero polynomial. The degree test compares \(\deg P\) and \(\deg Q\).

The criterion is simple, but it is important to apply it to the full polynomials. A numerator containing several terms does not have degree equal to its number of terms, and the degree of a product is found by adding the degrees of its polynomial factors.

This product rule applies when \(A\) and \(B\) are nonzero polynomials. In particular, a factored numerator or denominator can often be classified without expanding it.

Worked Examples: Comparing Degrees

Why Equal Degrees Still Require Division

It can be tempting to think that a numerator and denominator of equal degree are already suitable for partial fractions because neither degree is larger. The properness definition rules this out: the numerator must have strictly lower degree.

There is also a useful way to understand the equal-degree case. The leading terms of two degree-\(n\) polynomials are both multiples of \(x^n\). A constant multiple of the denominator can match and remove the numerator's leading term. That constant is the ratio of the leading coefficients. This is why the polynomial part produced by division in the equal-degree case is a constant, rather than a polynomial of positive degree.

The degree comparison determines whether division is needed; it does not require you to carry out that division. The full quotient-and-remainder method belongs to the next stage of the procedure.

Common Factors Do Not Change the Classification

A numerator and denominator may share a nonzero polynomial factor. Canceling such a factor reduces the degree of both polynomials by the same amount. Consequently, it does not change whether the numerator degree is less than, equal to, or greater than the denominator degree.

Here is the justification. Since \(P\) and \(Q\) are divisible by \(G\), their quotients are polynomials. The degree of a product of nonzero polynomials is the sum of the degrees of its factors. From \(P=G(P/G)\), it follows that \(\deg(P/G)=\deg P-\deg G\). Likewise, \(\deg(Q/G)=\deg Q-\deg G\). Subtracting the second equality from the first gives

The sign of this difference determines the classification: a negative difference means proper, while zero or a positive difference means improper. Because cancellation leaves the difference unchanged, it cannot switch a rational function from one class to the other.

Cancellation does have a separate domain issue. The original expression in the example is undefined at \(x=2\), even though the reduced formula \((3x+1)/(x+4)\) has a value there. Canceling a factor gives the same values wherever the original expression is defined, but it does not automatically restore points excluded from the original domain. This distinction matters when discussing the function's domain; it does not affect the degree test.

A Reliable Decision Process

Use the following sequence whenever you encounter a rational function and need to know whether division comes first.

Common Mistake: Treating Equality as Proper

A frequent error is to use “less than or equal to” in the definition of a proper rational function. For example, someone might classify \(\frac{x^2+1}{4x^2-7}\) as proper because the numerator degree is not greater than the denominator degree.

The calculation \(2\leq2\) is true, but it is not the test for properness. Properness requires a strict inequality. The correct comparison is \(2<2\), which is false, so the rational function is improper. The correction is to remember that equality belongs with the improper case.

Another source of error is counting visible factors rather than adding their degrees. For example, \((x^2+1)(x-1)\) has two displayed factors, but its degree is \(2+1=3\), not \(2\). Use polynomial degrees, not the number of terms or factors, in the comparison.

Why the Test Matters

Partial-fraction methods are designed to handle proper rational terms. When the numerator degree is at least the denominator degree, the expression must first be separated into a polynomial part and a proper rational part. Recognizing this before beginning prevents an attempted decomposition from being incomplete.

The degree test is also efficient: it does not require finding roots, factoring every polynomial, or solving for partial-fraction coefficients. It is a preliminary decision based only on the highest powers. If common factors are visible, they may help reveal the reduced expression, but they cannot change the proper-or-improper outcome.