Pure Mathematics 2

Topic Brief: Factor and Remainder Theorems

Provenance (2026-09-29). Read in full for this pass: research/veridian/WMA12-verified-facts.md (§2's transcription of spec item 2.1; §5's archive-scope note — 20 series with a question paper and mark scheme reviewed, June 2019–January 2026, 17 with a published examiner report; §6.2 in full; §7's "verified across at least 4 series" line; and both 2026-09-14 factor-and-remainder- theorems.ts mark-scheme-bullet-coverage audit entries, including the second, independent pass that added the June 2019 Q6(b) and Jan 2020 Q3(b) citations). Also read in full: content/veridian/curriculum/wma12/factor-and-remainder-theorems.ts (every series citation in the file) and content/veridian/technique-hub/wma12/codex-deep-dive-factor-and-remainder-theorems.md (Part 5 and its consolidated ledger table) and signal-frequency-and-boundaries.md (this topic's row). content/veridian/question-bank/wma12.ts grepped for "factor" / "remainder" — zero matches, no worked past-paper item there for this topic. Not affiliated with or endorsed by Pearson Edexcel. Quotations below are short (under ~15 words) and attributed by name; no full question, mark-scheme extract, or examiner-report passage is reproduced at length.

#SIGNAL Topic Master Brief — Factor and Remainder Theorems (WMA12, spec 2.1)

#What this is, and isn't

signal-frequency-and-boundaries.md gives this topic one row citing three series; the codex deep-dive's own audit-log research separately turned up two more, never before pulled into a single frequency count. This document does that consolidation — one narrow reference restating no theory (that's the codex deep-dive's Parts 1–4) and no general exam craft (forge-exam-technique-guide.md) — and adds the "what to expect before opening the question" summary that neither existing document states as such.

#Real frequency: seen in 5 of 20 series checked

The archive has 20 series with a question paper and mark scheme read. This topic (spec 2.1) is individually confirmed tested, by named series and question, in 5 of them: June 2019, Jan 2020, Oct 2023, Jan 2024, June 2025. The facts bank's own §7 line — "verified across at least 4 series" — was written before the June 2019 and Jan 2020 citations were separately confirmed in this file's own 2026-09-14 audit log; 5 is that same floor, now counted explicitly rather than left as "at least 4."

#Every real series/question that has tested this topic

Series	Question	Marks	Real angle tested	Source
June 2019	Q6(b)	not stated in sources read	Divide out an already-given factor, then factorise the quotient; quotient is a repeated factor, $(3x-2)^2$	`WMA12-verified-facts.md` §9 audit log
Jan 2020	Q3(b)	not stated in sources read	Same shape as June 2019 Q6(b); final answer required as a product, not a list, with isw noted	§9 audit log; §4
Oct 2023	Q4	9 (2, 2, 5)	Two unknowns from one given factor + one stated remainder, both "show that," then simultaneous equations feeding a full factorisation; the "hence" instruction is mark-scheme-enforced	§6.2; §9 (two independent audits)
Jan 2024	Q1	3	Single unknown from one stated remainder — the accessible-opener shape	§6.2
June 2025	Q4	9	Confirmed present via FORGE's independently-verified per-question map; no further per-question detail extracted in any document read	`forge-exam-technique-guide.md`; `signal-frequency-and-boundaries.md`

#Real mark-tariff pattern

Confirmed totals span 3 to 9 marks, and the tariff tracks the question's own shape rather than drifting: the two named single-part, single-unknown instances (Jan 2024 Q1) sit at the bottom of that range, and the two named multi-part instances that continue into a full factorisation (Oct 2023 Q4; June 2025 Q4, by category) sit at 9. Where the mark scheme is known in structural detail (Jan 2024 Q1; Oct 2023 Q4), the pattern is a short M1–M1–A1 or M1–A1 chain for the substitution-and-solve step, followed — only when a factorisation is asked for — by a dependent (dM1) factorising step tagged explicitly onto "their" quotient, never scored as a standalone method mark. June 2019 Q6(b) and Jan 2020 Q3(b) confirm that same dM1-dependent structure independently, on two further series, even though their overall totals were not stated in any source read.

#Real recurring context/angle across the series checked

Two patterns recur across more than one named series, not one:

Given-factor → divide → factorise-the-quotient recurs with an identical mark structure on two separate series. June 2019 Q6(b) and Jan 2020 Q3(b) both tag the quotient-factorising step dependent, not independent — the same dM1 convention, confirmed twice rather than once.
Where a full worked-example error record exists (Oct 2023 Q4), the dominant error is a sign mismatch between which bracket form is given and which root gets substituted — recorded in both directions on the same question at once. Jan 2024 Q1's own record shows the analogous slip on a simpler, single-unknown version of the same question shape.

One finding is confirmed on a single series but is mark-scheme-mandated rather than merely observed, so it belongs here regardless of count: Oct 2023 Q4's scheme states a "hence factorise" instruction forfeits the entire part's marks if the given factor is bypassed, "even" when the final answer is completely correct.

#If this topic appears: what to expect

Expect a standalone, spec-2.1 question, most often opening the paper's algebra content or appearing as a self-contained multi-part item — confirmed totals run from 3 marks (a single stated remainder) up to 9 (two conditions feeding a full factorisation).
If a factor is already given, expect the marking to start at the division itself — no antecedent mark exists in either named anchor for re-confirming a factor the question already states.
Expect the step "factorise the quotient you just divided out" to be marked dependent on the division above, on every series where the structure is known — never a free-standing method mark.
If the instruction says "hence," expect that word to be enforced: a correct final factorisation reached without visibly using the given factor is confirmed, on a real mark scheme, to score zero for that part.
Expect the single most commonly recorded slip to be substituting the wrong sign for the value that zeroes the divisor — confirmed in both directions on the one series with full error detail.

#What this document deliberately leaves out

No derivation of the remainder/factor identity itself, no worked search-from-scratch mechanism, and no two-unknown method-choice framework — all in codex-deep-dive-factor-and-remainder-theorems.md, Parts 1–4. No general exam-timing or self-checklist material (forge-exam-technique-guide.md). No percentage, likelihood, or grade-boundary claim of any kind, per this course's standing discipline — paper-wide grade boundaries live in signal-frequency-and-boundaries.md Part 2, and none there is topic-specific.

#Sources used

research/veridian/WMA12-verified-facts.md — §2, §5, §6.2, §7, and both 2026-09-14 factor-and-remainder-theorems.ts audit-log entries, read in full.
content/veridian/curriculum/wma12/factor-and-remainder-theorems.ts — read in full for its own series citations.
codex-deep-dive-factor-and-remainder-theorems.md (Part 5 and its consolidated ledger table), signal-frequency-and-boundaries.md (this topic's row) — read to confirm nothing above restates them.
content/veridian/question-bank/wma12.ts — grepped for "factor" and "remainder"; zero matches.