Proper rational expression

1 Definition and Degree Condition

1.1 General form of rational expressions

A rational expression is a quotient of two polynomials, typically written as \[ \frac{P(x)}{Q(x)}, \quad Q(x)\neq 0. \] The condition \(Q(x)\neq 0\) specifies where the expression is defined. In algebraic manipulations, rational expressions are treated symbolically, but domain restrictions arise from the denominator.

1.2 The “proper” criterion (\(\deg(P)<\deg(Q)\))

A proper rational expression is one in which the degree of the numerator is strictly smaller than the degree of the denominator: \[ \deg(P) < \deg(Q). \]

1.3 Examples of proper and non-proper cases

• Proper example: \(\dfrac{x^2+1}{x^3-4x}\). Here \(\deg(P)=2\) and \(\deg(Q)=3\), so the expression is proper.
• Non-proper example: \(\dfrac{x^3+2}{x^2-1}\). Now \(\deg(P)=3\) and \(\deg(Q)=2\), so it is improper.
• Borderline case (not proper): \(\dfrac{2x^2+1}{x^2-3}\). Since degrees are equal, \(\deg(P)=\deg(Q)\), the expression is not proper.
• Constant numerator: \(\dfrac{5}{x^2+1}\) is proper because the numerator has degree \(0\) (for a nonzero constant) and the denominator has degree \(2\).

2 Properties of Proper Rational Expressions

2.1 Domain considerations and restrictions (\(Q(x)\neq 0\))

For \(\dfrac{P(x)}{Q(x)}\), the expression is undefined wherever \(Q(x)=0\). If the numerator shares factors with the denominator, algebraic simplification may remove some restrictions, but only after cancellation is performed carefully. After simplification, the remaining denominator determines the actual domain.

2.2 Behavior at infinity (end behavior using degrees)

2.3 Simplification by factoring

Factoring polynomials often reveals structure needed for simplification and decomposition. For example, if \[ Q(x)=(x-a)^2(x^2+bx+c), \] then the factors dictate the form of partial fractions used later. Factoring also helps identify removable factors that might cancel with the numerator, changing the effective degrees and leading to a proper reduced form.

2.4 Cancellation and preserving properness

If a common factor \(R(x)\) divides both numerator and denominator, one may cancel it: \[ \frac{P(x)}{Q(x)}=\frac{\frac{P(x)}{R(x)}}{\frac{Q(x)}{R(x)}} \quad \text{(with updated restrictions)}. \] Cancellation can preserve properness because dividing both polynomials by the same factor reduces degrees by equal amounts. However, the domain must be updated: values that originally made \(Q(x)=0\) but also caused cancellation may still be excluded unless the simplified form is defined there.

3 Algebraic Manipulation Techniques

3.1 Adding, subtracting, and common denominators

To add or subtract rational expressions, a common denominator is typically chosen: \[ \frac{P_1(x)}{Q_1(x)} \pm \frac{P_2(x)}{Q_2(x)} = \frac{P_1(x)Q_2(x) \pm P_2(x)Q_1(x)}{Q_1(x)Q_2(x)}. \] The resulting expression may be proper or improper depending on the degrees. After combining, factoring and degree checks help restore a proper form when desired.

3.2 Multiplying rational expressions

Multiplication is straightforward: \[ \frac{P_1(x)}{Q_1(x)}\cdot \frac{P_2(x)}{Q_2(x)}=\frac{P_1(x)P_2(x)}{Q_1(x)Q_2(x)}. \] Properness may or may not hold after multiplication. The degree of the product numerator is \(\deg(P_1)+\deg(P_2)\), and the degree of the product denominator is \(\deg(Q_1)+\deg(Q_2)\); properness holds if the numerator degree stays strictly smaller.

3.3 Dividing rational expressions

Division by a nonzero rational expression follows: \[ \frac{P_1(x)}{Q_1(x)}\div \frac{P_2(x)}{Q_2(x)} = \frac{P_1(x)Q_2(x)}{Q_1(x)P_2(x)}. \] Again, domain restrictions expand: the new denominator cannot be zero, so zeros of \(Q_1(x)\) and \(P_2(x)\) must be excluded.

3.4 Checking properness after operations

After any algebraic operation, one can verify properness by comparing degrees of the resulting numerator and denominator. If improper, the expression can be converted to a polynomial plus a proper rational remainder (see Chapter 5). This step is particularly important before partial fraction decomposition, which assumes a proper rational part.

4 Partial Fraction Decomposition (Core Application)

4.1 When decomposition applies

Partial fraction decomposition is most convenient for rational expressions where the fraction is proper. A general rational expression can be separated into:

1. a polynomial part (from converting improper fractions), and
2. a proper rational fraction that can be decomposed into simpler terms.

Once in proper form, the denominator’s factorization determines the decomposition template.

4.2 Factorization of the denominator

The denominator \(Q(x)\) is factored into linear and irreducible quadratic factors over the relevant number system. For real-coefficient problems, quadratic factors irreducible over the reals are treated as blocks. For example, \[ Q(x)=(x-a)^m(x-b)^n\cdot (\text{irreducible quadratic})^k \] leads to corresponding families of terms in the decomposition.

4.3 Decomposition with linear factors

If the denominator contains a linear factor \((x-a)^m\), the contribution typically has the form \[ \frac{A_1}{x-a}+\frac{A_2}{(x-a)^2}+\cdots+\frac{A_m}{(x-a)^m}. \] These constants \(A_i\) are chosen so that, when everything is combined over the original denominator, the identity holds for all \(x\) where the expression is defined.

4.4 Decomposition with irreducible quadratic factors

If the denominator includes an irreducible quadratic factor \((x^2+bx+c)^m\), the corresponding terms look like \[ \frac{A_1x+B_1}{x^2+bx+c}+\frac{A_2x+B_2}{(x^2+bx+c)^2}+\cdots+\frac{A_mx+B_m}{(x^2+bx+c)^m}. \] Each numerator is linear in \(x\) to match the degree needed for recombination.

4.5 Recombining and verifying results

Verification is done by recomposition: multiply the proposed decomposition by the original denominator and simplify. The resulting identity should match the original numerator polynomial. This check is essential because it confirms both coefficients and the correct decomposition structure, including multiplicities of repeated factors.

5 Converting Improper to Proper Forms

5.1 Long division for rational expressions

When \(\deg(P)\ge \deg(Q)\), one performs polynomial long division: \[ \frac{P(x)}{Q(x)} = S(x) + \frac{R(x)}{Q(x)}, \] where \(S(x)\) is a polynomial and \(\deg(R)<\deg(Q)\). The fraction \(\dfrac{R(x)}{Q(x)}\) is then proper and suitable for partial fraction decomposition.

5.2 Polynomial part plus proper remainder

The separation into a polynomial part and a proper remainder is systematic: the division removes the high-degree behavior carried by the numerator. The remainder \(R(x)\) contains only terms that cannot be further reduced by dividing by \(Q(x)\) in polynomial form.

5.3 Degree checks after division

After obtaining \(R(x)\), a degree check confirms properness: \[ \deg(R)<\deg(Q). \] If \(R(x)=0\), the expression reduces exactly to the polynomial \(S(x)\) with no proper fractional part. This situation affects decomposition because there is nothing further to split.

5.4 Edge cases (constant numerator, equal degrees)

• Equal degrees: If \(\deg(P)=\deg(Q)\), the division yields a nonzero constant polynomial part plus a proper remainder of lower degree.
• Constant nonzero numerator: If \(P(x)\) is constant, then \(\deg(P)=0\). The expression is proper whenever \(\deg(Q)\ge 1\), so conversion may not be required.

6 Common Worked Examples

6.1 Simple proper fractions

Consider \[ \frac{3}{x^2-1}. \] Factor the denominator: \[ x^2-1=(x-1)(x+1). \] The decomposition has the form \[ \frac{3}{(x-1)(x+1)}=\frac{A}{x-1}+\frac{B}{x+1}. \] Solving for \(A\) and \(B\) by recombination yields a correct identity and confirms that the original is already proper (numerator degree \(0\), denominator degree \(2\)).

6.2 Proper expressions requiring decomposition

Take \[ \frac{x+2}{x(x-3)}. \] This is proper because \(\deg(x+2)=1&lt;\deg(x(x-3))=2\). Decompose using linear factors: \[ \frac{x+2}{x(x-3)}=\frac{A}{x}+\frac{B}{x-3}. \] Multiplying through by \(x(x-3)\) gives \[ x+2=A(x-3)+Bx. \] Matching coefficients (or substituting convenient values of \(x\)) determines \(A\) and \(B\).

6.3 Simplifying before decomposing

Suppose \[ \frac{2(x-1)}{(x-1)(x+3)}. \] Before decomposition, cancel \((x-1)\) (noting the original restriction \(x\ne 1\)): \[ \frac{2(x-1)}{(x-1)(x+3)}=\frac{2}{x+3}. \] This step avoids unnecessary work and yields an already-simple proper rational expression.

6.4 Verification by recomposition

For any proposed decomposition, verification is performed by recomposition. If one claims \[ \frac{P(x)}{Q(x)}=\sum_i \frac{N_i(x)}{D_i(x)}, \] then multiplying both sides by \(Q(x)\) and simplifying should reproduce \(P(x)\) exactly. If the identity fails, the coefficients in the partial fractions must be corrected; if it succeeds, the decomposition is confirmed.