Deep Dive: Partial Fractions Decomposition and Integration

Provenance: written 2026-09-29. Read/checked this session: research/veridian/WMA14-verified-facts.md §1 (spec text for 2.1, 4.1, 6.3), §3 (formula-booklet contents — confirmed no discrete partial-fractions formula exists there; the general-$n$ binomial series that 4.1 leans on is booklet-given), §5.2 (this topic's own three real citations), §6 (series-reviewed scope), §7 (the topics-with-enough-material list — this topic is not on it), §8 (evidence-strength flags); the existing lesson file content/veridian/curriculum/wma14/partial-fractions-decomposition-and-integration.ts in full (every mechanism block, worked example, method-comparison, trap-taxonomy entry and MCQ, plus its own header and footer provenance notes); content/veridian/technique-hub/wma14/codex-formula-and-method-bank.md §2 and §2.2 (formula-status and method-choice skeleton) and its §6 integration entry; forge-exam-technique-guide.md (the algebraic-manipulation checklist and cross-topic trap-taxonomy row); signal-frequency-and-boundaries.md §1 (the frequency row: 3 of 7 series, no confirmed total mark size); content/veridian/question-bank/wma14.ts (checked — no item for this spec point yet); and this technique-hub's own codex-deep-dive-implicit-and-parametric-differentiation-tangents-and-normals.md, read as the format precedent for this document type. No new primary Pearson document was fetched this session — every exam citation below traces to the facts bank's own already-verified §5.2 quotes; the new material this document adds (§1's uniqueness proof, §1.4's account of why quadratic factors are excluded, and §3.3's differentiation worked example) is original mathematical/pedagogical exposition grounded directly in the spec's own text, labelled as such throughout rather than presented as exam-sourced. Not affiliated with or endorsed by Pearson Edexcel.

#WMA14 · CODEX Deep-Dive — Partial Fractions Decomposition and Integration

#Why this document exists, and what it does not repeat

The lesson file already contains a full worked treatment of every sub-case below: three distinct linear factors, one repeated linear factor, a top-heavy numerator, the $\frac1a$ integration rule, and the spec-4.1 series-expansion extension — each with its own mechanism block, marked solution, method comparison and MCQ, all built from real citations. This document does not re-run any of that. It exists for three things the lesson, the formula bank and the FORGE checklist do not do:

1. Prove, rather than assert, why the decomposition is unique — the lesson's own mechanism block explains why substitution and coefficient-comparison are both valid (§1.1 below restates that in one line and cites it), but nowhere in this course is there a direct proof that they must always agree.
2. Explain the mechanism behind the one sub-case the spec explicitly excludes — an irreducible quadratic factor — rather than only citing that it is excluded (§1.4).
3. Cover the one application spec 2.1's own guidance names that has zero coverage anywhere else in this course: differentiation. Integration (spec 6.3) and series expansion (spec 4.1) each already have a full worked treatment; differentiation has none, in the lesson, the formula bank or the FORGE guide, despite being named in the same guidance sentence as the other two (§3.3).

Scope boundary, confirmed against the facts bank rather than assumed: this document stays inside spec 2.1 (decomposition) and 6.3 (the integration it feeds), the same two points the lesson file covers. It does not extend into spec 4.1's binomial-series machinery beyond citing where the lesson already connects the two.

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#1. The theoretical model

#1.1 What licenses the two solving methods (restated in one line, not re-derived)

A partial-fraction decomposition is asserted with $\equiv$, not $=$: a claim that two expressions are identical for every value of $x$, not merely equal for some. That single fact is the entire justification for both cover-up substitution (safe to evaluate the identity at any $x$, including a root of the original denominator) and coefficient comparison (two polynomials identical for every $x$ must have identical coefficients). The lesson file's own mechanism block derives this in full; it is not repeated here.

#1.2 Why a decomposition exists at all: a counting argument

For a denominator built from $k$ distinct linear factors (degree $k$), spec 2.1 requires a proper numerator — degree at most $k-1$, i.e. $k$ independent coefficients to match ($x^{k-1}$ down to $x^0$). Assume the decomposition $\sum_i \frac{A_i}{(\text{factor}_i)}$ and clear denominators: the numerator becomes $\sum_i A_i\times(\text{product of every OTHER factor})$, a polynomial of degree exactly $k-1$ carrying $k$ unknown constants $A_i$. Equations and unknowns match exactly — $k$ coefficients to satisfy, $k$ free constants to satisfy them with. This is the same counting logic the lesson file's own mechanism block already applies to the repeated-factor case (there, a degree-3 denominator with one repeated factor needs three unknowns to match three coefficients, which is why a single term for the repeated factor is never enough) — restated here at the level of the general distinct-factor case it generalises, rather than re-derived for the specific repeated case the lesson already owns.

#1.3 Why the decomposition, once it exists, is unique — a direct proof

Existence (§1.2) shows the counting works out; it does not by itself show two different candidate decompositions can't both be correct. This is checkable directly, and is genuine additive content — no document in this course currently proves it. Suppose two decompositions of the same proper fraction over $k$ distinct linear factors both held:

$$\sum_i \frac{A_i}{x-r_i} = \sum_i \frac{A_i'}{x-r_i}.$$

Subtracting and clearing denominators gives a polynomial identity, true for every $x$:

$$\sum_i (A_i-A_i')\times\!\!\prod_{j\neq i}(x-r_j) = 0 \quad\text{for all } x.$$

Evaluate this at $x=r_1$: every term in the sum except the $i=1$ term contains the factor $(x-r_1)$ explicitly and vanishes; the surviving term is $(A_1-A_1')\times\prod_{j\neq1}(r_1-r_j)$, and that product is nonzero precisely because the $k$ roots are distinct. A product equal to zero with a nonzero factor forces the other factor to be zero: $A_1=A_1'$. The same substitution at $r_2,\dots,r_k$ in turn forces every remaining constant to match. This is exactly why the cover-up rule always produces the unique right answer, rather than one of several possible answers: distinctness of the roots is not a convenience for the method, it is the fact that makes the decomposition unique in the first place. For a repeated root the same argument extends (evaluate the cleared identity's own derivative at the repeated root to isolate the next constant, rather than the identity itself a second time) but is not spelled out in full here, since the counting argument in §1.2 and the lesson's own repeated-factor mechanism block already establish the practical version a P4 student needs.

#1.4 Why an irreducible quadratic factor is genuinely excluded, not arbitrarily excluded

Spec 2.1's guidance names the boundary directly: a denominator factor of the shape $(x^2+a)$ with $a>0$ is not required on this paper. The lesson file cites this exclusion and shows the one worked contrast ($\frac{x}{x^2+5}$, handled by recognition rather than decomposition); neither it nor the formula bank states the mathematical reason the boundary sits exactly there. It is this: cover-up substitution needs a real root of the factor being isolated — a value of $x$ that sends that one bracket to zero while every other bracket in the identity survives. A linear factor $ax+b$ always has exactly one, at $x=-\frac{b}{a}$. A quadratic factor $x^2+a$ with $a>0$ has discriminant $-4a<0$: no real root exists, so there is no real value of $x$ that isolates a constant sitting over it the way cover-up isolates every other constant in this topic. The general algebraic theory does have a matching form for this case — a linear numerator over the quadratic factor, $\frac{Px+Q}{x^2+a}$, two unknowns rather than one — but solving for $P$ and $Q$ needs coefficient-comparison or complex substitution, neither of which is P4 content, which is exactly why the spec draws the line where it does: every technique this paper actually teaches (cover-up by a real root, or coefficient-comparison against a linear/repeated-linear assumed form) depends on the denominator factoring completely over the reals into linear pieces. $x^2+5$ does not, so it was never a partial-fractions question at all — it is handled by recognition instead, precisely because no real factorisation exists to decompose.

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#2. The three required sub-cases — restated compactly, each with a short original micro-example

Full worked chains, marked solutions and trap-taxonomy for all three already exist in the lesson file; the micro-examples below use different numbers from every one of the lesson's own worked examples, specifically so nothing here could be mistaken for a reproduction of that file's content.

#2.1 Three distinct linear factors

Spec 2.1 names the shape $(ax+b)(cx+d)(ex+f)$ directly. Micro-example: $\dfrac{2x^2+5x-1}{(x-1)(x+1)(x+2)}$ decomposes, by cover-up at $x=1$, $x=-1$, $x=-2$ in turn, to

$$\frac{1}{x-1}+\frac{2}{x+1}-\frac{1}{x+2}$$

(verified by clearing denominators and confirming the recombined numerator matches term by term — every numeric claim in this document was checked this way before being written in, the same discipline the lesson file's own header states for its content). This specific decomposition is reused, not re-derived, in §3 below to build the "one decomposition, three applications" synthesis.

#2.2 One repeated linear factor

Spec 2.1's other named shape, $(ax+b)(cx+d)^2$. Micro-example: $\dfrac{3x^2-6x+9}{(x+1)(x-2)^2}$ decomposes to

$$\frac{2}{x+1}+\frac{1}{x-2}+\frac{3}{(x-2)^2}.$$

The repeated root $x=2$ isolates the highest-power constant (here, 3) directly by cover-up in one substitution, exactly as the lesson's own method-comparison demonstrates; the middle constant (1) still needs either a second convenient value or coefficient comparison, for the counting reason given in §1.2.

#2.3 Numerator degree $\geq$ denominator degree

Spec 2.1 states this is a required case, not an edge case. Micro-example, degree of numerator equal to degree of denominator: $\dfrac{x^2+x-1}{(x-1)(x+2)}$. Dividing $x^2+x-1$ by $(x-1)(x+2)=x^2+x-2$ gives a quotient of $1$ and a remainder of $1$ (since $(x^2+x-1)-(x^2+x-2)=1$), so

$$\frac{x^2+x-1}{(x-1)(x+2)} = 1+\frac{1}{(x-1)(x+2)} = 1+\frac{1}{3(x-1)}-\frac{1}{3(x+2)}.$$

The lesson file's own worked example for this sub-case uses long division and the identity method side by side on a different numerator; this micro-example exists only to keep §2's three sub-cases parallel in one place, not to re-run that comparison.

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#3. One decomposition, three applications — the synthesis spec 2.1's own guidance asks for directly

Spec 2.1's guidance closes with one sentence naming all three uses this decomposition is ever put to on this paper: integration, differentiation, series expansion. The lesson file and formula bank each cover integration and series expansion — with different worked examples for each, never the same one run through all three. This section reuses §2.1's own decomposition,

$$f(x)=\frac{2x^2+5x-1}{(x-1)(x+1)(x+2)} \equiv \frac{1}{x-1}+\frac{2}{x+1}-\frac{1}{x+2},$$

for all three, which is the genuinely new structural contribution this section makes: showing that one decomposition is the shared starting point spec 2.1's guidance sentence is actually describing, not three unrelated skills that happen to share a name.

#3.1 Integration (cited, not re-derived)

$$\int f(x)\,dx = \ln|x-1| + 2\ln|x+1| - \ln|x+2| + c.$$

Every $\frac1a$ scaling factor here is trivially $1$ (each bracket's $x$-coefficient already is), so this particular decomposition does not itself exercise the two-series-confirmed $\frac1a$-omission slip the lesson's own marked solution is built around (Oct 2022, Q2; Jan 2024, Q2(b) — see §4). It is included here only to complete the three-application picture; the lesson file remains the place for the fully worked, error-anchored integration treatment.

#3.2 Series expansion (cited, not re-derived)

Expanding $f(x)$ in ascending powers of $x$ about $x=0$ uses the same $(1+u)^n$ machinery the lesson's own spec-4.1 block already drills in full (factor a constant from each bracket, apply the general binomial series, sum the three resulting series, take the tightest of the three individual validity conditions). For this decomposition the individual conditions are $|x|<1$ (from the $x-1$ factor), $|x|<1$ (from the $x+1$ factor) and $|x|<2$ (from the $x+2$ factor), so the combined range is $|x|<1$ — the same "most-restrictive-of-the-three" rule the lesson's own worked chain already establishes and is not re-derived here.

#3.3 Differentiation — the genuine content gap this document fills

No document in this course currently works this application, despite spec 2.1 naming it in the same breath as the other two. There is no examiner-report or mark-scheme citation for it anywhere in the facts bank either — this is stated plainly rather than dressed up as exam-sourced, per this course's own no-fabrication discipline (§5 below repeats the point explicitly).

Differentiating the decomposed form is direct chain-rule work on three single-power terms, with no quotient rule required at all:

$$f'(x) = -\frac{1}{(x-1)^2} - \frac{2}{(x+1)^2} + \frac{1}{(x+2)^2}.$$

Differentiating the original quotient form directly would instead require the quotient rule (or the product rule on a factored denominator raised to $-1$) on a degree-2-over-degree-3 rational function in one un-split step — mechanically riskier, with more terms to track in a single application, for the identical result. This is the real content of "partial fractions as an application to differentiation": decomposition does not just simplify the algebra of finding an antiderivative, it does the same simplifying work for a derivative, turning one compound-quotient differentiation into three short chain-rule applications, each of the same shape as $\frac{d}{dx}\!\left[(x-a)^{-1}\right]=-(x-a)^{-2}$. A genuine use for this beyond the mechanical saving: locating where $f'(x)=0$ (a stationary point of the original rational function) is a three-term equation in the decomposed form rather than a single quotient-rule numerator that would otherwise need expanding out to a common denominator first — the decomposition is doing real algebraic work before any calculus-specific step happens at all.

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#4. Synthesis — every real exam angle this specific topic has actually been tested from

Restricted strictly to spec 2.1/6.3, the facts bank's question-level review yields exactly three real, citable angles, across three series — confirmed as 3 of 7 series that received real question-by-question mining, per this course's own SIGNAL document, with no single confirmed total mark size captured for a partial-fractions question in either source document.

What this table adds beyond SIGNAL's frequency row and FORGE's checklist: those two documents record that three angles exist and list the trap-taxonomy names for each. This table is the first place in the technique-hub layer that maps each real angle onto the specific sub-case (§2.1–§2.3) it exercises — the same connective role this technique-hub's implicit/parametric-differentiation deep-dive already establishes for its own topic.

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#5. What is genuinely not evidenced — stated rather than invented

• The differentiation application (§3.3) has zero exam-report or mark-scheme citation anywhere in the facts bank. It is spec-guidance-named, and the worked treatment above is real, checked mathematics — but it is VERIDIAN pedagogical exposition of a named spec application, not a documented exam angle, and is not represented as one anywhere above.
• The repeated-factor power-rule integral ($\int\frac{1}{(ax+b)^n}dx$ for $n\geq2$, e.g. the spec-named $3/(x-1)^2$) has no citation confirming a specific tested error for it — every real, quoted error in §4 concerns the ln-form $\frac1a$ factor, never the power-rule form. The power-rule integral is spec-required (named directly in 6.3's guidance) but its exam-error record, if any, was not found in this course's review.
• No confirmed total mark size exists for a whole partial-fractions question anywhere in either source document, per SIGNAL's own explicit "—" entry for this row; none is invented here.
• No grade-boundary figure exists anywhere in this course's WMA14 source review, for this or any topic — restated here, as every other technique-hub document in this course restates it, so that a deep-dive is never the place a fabricated boundary or likelihood percentage quietly appears.
• The quadratic-factor case (§1.4) is out of scope by the spec's own text, not under-evidenced — it is not a thin-evidence gap, it is a genuine boundary, and nothing above treats it as a topic this paper tests.

This is the honest shape of the evidence for this topic: three real citations across three series, a counting/uniqueness argument and an exclusion mechanism that are genuine mathematics rather than exam citations, and one spec-named application with no documented exam angle at all. The facts bank's own §7 does not list this topic among those with "enough verified material for a genuine standalone lesson" — consistent with why this document is the length it is rather than longer, and why §5 is as explicit as it is about where the real record stops.

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#Cross-references (cited, not duplicated)

• Full worked chains, marked solutions, method-comparisons, trap-taxonomy and MCQs for every sub-case in §2 and the integration application in §3.1: content/veridian/curriculum/wma14/partial-fractions-decomposition-and-integration.ts.
• The formula-status table and five-line method-choice skeleton this document's §1–§2 sit alongside: codex-formula-and-method-bank.md §2, §2.2.
• The algebraic-manipulation checklist and trap-taxonomy name list: forge-exam-technique-guide.md.
• The frequency-of-appearance figure used in §4: signal-frequency-and-boundaries.md §1.
• The full spec-4.1 series-expansion worked chain cited in §3.2: content/veridian/curriculum/wma14/partial-fractions-decomposition-and-integration.ts (its own SPEC_4_1-tagged block).

#Sources used

research/veridian/WMA14-verified-facts.md §1 (spec 2.1/4.1/6.3 verbatim text), §3 (formula-booklet contents), §5.2 (the three real citations synthesised in §4: Oct 2021 Q3(a), Oct 2022 Q2, Jan 2024 Q2(a)–(b)), §6 (series-reviewed scope, used for §4's "3 of 7" figure via SIGNAL's own re-derivation), §7 (confirms this topic is not on the "enough material for a standalone lesson" list), §8 (evidence-strength flags). signal-frequency-and-boundaries.md §1 (the frequency-table row for this topic, re-cited exactly, not re-derived). No new primary Pearson document was fetched this session; §1.2–§1.4 and §3.3 are original mathematical/pedagogical exposition grounded in the spec's own text rather than in an exam citation, and are labelled as such at each point they appear.