Why Recombine a Proposed Decomposition?
A partial-fraction decomposition can look plausible while still containing an incorrect coefficient, a missing term, or an algebra error. Recombining the terms gives a direct check: place them over a common denominator, simplify the numerator, and compare it with the numerator that the decomposition is supposed to represent.
This check is especially useful after solving a coefficient system or using a shortcut. It does not require repeating the method used to find the coefficients. Instead, it tests the proposed result by reversing the decomposition step.
The Basic Check
Suppose a proper rational function has denominator \(D(x)\), and a proposed decomposition has terms with denominators that divide \(D(x)\). To check the proposal, multiply each term's numerator by the missing factors needed to give it denominator \(D(x)\). Add these adjusted numerators. The sum must equal the target numerator as a polynomial.
Here the least common denominator is chosen so that each quotient \(D(x)/D_i(x)\) is a polynomial. If the target is \(P(x)/D(x)\), the comparison is between \(P(x)\) and the numerator formed by adding the adjusted numerators. This comparison must be a polynomial identity: it must hold for every \(x\), not merely for a few selected values.
For an improper rational function, first distinguish its polynomial quotient from its proper remainder. If division gives \(P(x)=Q(x)D(x)+R(x)\), then the partial fractions represent \(R(x)/D(x)\), not \(P(x)/D(x)\) by itself. The recombined numerator of the partial fractions should match \(R(x)\).
Why the Check Works
The justification is the quotient-remainder identity together with ordinary fraction arithmetic. Suppose a proposed decomposition for a proper remainder is \(R(x)/D(x)\), where \(D(x)\) is also the common denominator of its terms. Combining those terms gives \(S(x)/D(x)\) for some polynomial \(S(x)\). The proposal is correct precisely when \(R(x)/D(x)=S(x)/D(x)\) wherever \(D(x)\ne0\).
Multiplying both sides by the nonzero denominator gives the numerator equality. Conversely, if the numerator polynomials are equal, dividing by \(D(x)\) wherever it is nonzero recovers equality of the rational expressions. Thus recombination checks the decomposition directly; it does not depend on how its coefficients were obtained.
Worked Example: Two Distinct Linear Factors
Worked Example: Checking a Decomposition with Distinct Factors
Check the proposed decomposition
The common denominator is \((x-1)(x+2)\). The first term needs the factor \(x+2\), and the second needs the factor \(x-1\). Therefore the combined numerator is
The combined denominator remains \((x-1)(x+2)\), so recombination gives
This is the target rational expression, so the proposed decomposition is verified for \(x\ne1\) and \(x\ne-2\). The excluded values are the zeros of the denominator; neither side is defined there.
Worked Example: A Repeated Linear Factor
Worked Example: Checking Terms with a Repeated Factor
Check the proposed decomposition
Use the complete denominator \((x-1)^2(x+2)\). The term with denominator \(x-1\) needs the factor \((x-1)(x+2)\). The term with denominator \((x-1)^2\) needs \(x+2\). The term with denominator \(x+2\) needs \((x-1)^2\). The resulting numerator is
Thus the right-hand side recombines to
which matches the target. The decomposition is verified wherever \(x\ne1\) and \(x\ne-2\). Notice that each repeated power contributes its own term to the common-denominator calculation; the squared factor cannot be treated as though it were just \(x-1\).
Checking an Improper Rational Function
For an improper rational function, the numerator degree is at least the denominator degree. The earlier material on division expresses such a function as a polynomial quotient plus a proper rational remainder. A proposed partial-fraction decomposition applies to that proper remainder. A full check therefore has two parts: confirm the remainder decomposition, then combine the quotient with the remainder over the original denominator.
Worked Example: Checking a Decomposition After Division
Check the proposed identity
First verify the division step. Multiplying the proposed quotient by the denominator gives
Subtracting this product from the original numerator leaves the remainder
Now check the proposed partial fractions for this remainder. Since \(x^2-1=(x-1)(x+1)\), put the terms over that denominator:
The recombined numerator is exactly the proper remainder \(3x+7\). Finally, combine the quotient and the remainder fraction over \(x^2-1\):
This reproduces the original rational function. The identity is verified for \(x\ne1\) and \(x\ne-1\), the values where the denominator vanishes.
A Common Pitfall: Forgetting the Missing Factors
When adding partial fractions, each numerator must be multiplied by the factors missing from that term's denominator. Adding the visible numerators alone is not a valid common-denominator calculation.
Worked Example: Correcting a Missing-Factor Error
Consider the proposed expression \(2/(x-1)+3/(x+2)\). A mistaken check might claim that its numerator is simply \(2+3=5\), and therefore write
The common denominator is \((x-1)(x+2)\), but the numerators must be adjusted as well. The correct calculation is
The incorrect numerator \(5\) arose from failing to multiply each term by its missing factor. The correction is to write the common denominator first and then construct every adjusted numerator explicitly.
A Reliable Recombination Routine
Use the following sequence to check a proposed decomposition. Keeping the target numerator visible throughout helps prevent confusion between an original numerator and a proper remainder.
Write down whether the terms should represent the whole rational function or only its proper remainder after division.
Use the full denominator required by the proposed terms, including all repeated powers.
Multiply each term's numerator by the factors missing from its denominator.
Expand carefully, combine like powers of \(x\), and compare the resulting polynomial with the target numerator or proper remainder.
The rational identity holds where the common denominator is nonzero.
Substituting a few convenient values can help locate an error, but it cannot replace the polynomial comparison. Two different polynomials may agree at several selected values and disagree elsewhere. A complete verification shows that the expanded numerator has exactly the same coefficients for every power of \(x\) as the target numerator.
What Recombination Establishes
A successful check establishes that the proposed terms add up to the stated rational function on its domain. It can confirm a decomposition even when the coefficients were obtained by a different route. A failed check identifies that something in the proposal or its algebra must be corrected, although it does not by itself tell which coefficient is wrong.
Recombination is also a useful way to distinguish two separate tasks. Finding a decomposition is the task of determining terms that represent the proper remainder. Checking it is the task of demonstrating, by a common-denominator calculation, that those terms really do represent that remainder. The latter should be done even when the proposed coefficients seem obvious.
Check Your Understanding
For each question, focus on the common denominator and on which numerator the result must match.
- When checking \(A/(x-1)+B/(x+2)\), what factors multiply \(A\) and \(B\) when the terms are combined?
- Why must the full repeated factor \((x-1)^2\) appear in the common denominator when checking terms with denominators \(x-1\), \((x-1)^2\), and \(x+2\)?
- If division gives \(P(x)=Q(x)D(x)+R(x)\), which polynomial should the recombined partial fractions match?
- Why does checking equality at a few selected values not establish that two numerator polynomials are identical?
- For what \(x\)-values is a rational identity with denominator \((x-1)(x+2)\) defined?
