---
title: "A-Maths Polynomials & Partial Fractions Guide"
description: "Polynomials and partial fractions reward a methodical approach. This guide covers the remainder and factor theorems, solving cubics, and the three partial-fraction cases."
author: "Gabriel"
author_url: "https://ancourage.academy/authors/gabriel"
published_at: 2026-07-13
modified_at: 2026-07-13
category: "teaching"
tags: ["Mathematics", "Secondary", "O-Level", "A-Maths", "SEC", "Polynomials", "Partial Fractions", "Singapore", "Exam Tips"]
canonical: "https://ancourage.academy/articles/a-maths-polynomials-partial-fractions-singapore"
source: "https://ancourage.academy/articles/a-maths-polynomials-partial-fractions-singapore"
language: "en-SG"
word_count: 1453
reading_time: "PT8M"
cover_image: "https://ancourage.academy/academic-pic/IMG_0142.jpg"
reviewed_by: "Min Hui"
---

# A-Maths Polynomials & Partial Fractions Guide

Polynomials and partial fractions reward a methodical approach. This guide covers the remainder and factor theorems, solving cubics, and the three partial-fraction cases.

**Polynomials and partial fractions are among the most method-driven topics in A-Maths — the remainder and factor theorems crack almost every cubic, and partial fractions follow three fixed templates depending on the denominator.** Students rarely struggle with the ideas; they struggle when they skip the structured steps. This guide is from [Ancourage Academy](https://ancourage.academy/academy), whose [secondary A-Maths tuition](https://ancourage.academy/courses/academy/secondary/a-maths) teaches these procedures step by step in small groups of 3–6 at [Bishan](https://ancourage.academy/find-us/bishan) and [Woodlands](https://ancourage.academy/find-us/woodlands).

This is a single-topic deep-dive — a sibling to our [A-Maths quadratics](https://ancourage.academy/articles/a-maths-quadratic-functions-equations-inequalities-singapore) and [binomial theorem](https://ancourage.academy/articles/a-maths-binomial-theorem-expansion-guide-singapore) guides. If you are still deciding whether to take A-Maths, read [E-Maths vs A-Maths](https://ancourage.academy/articles/e-maths-vs-a-maths-difference-singapore) first.

**If cubic equations or partial fractions are a gap, Ancourage Academy's [Sec 4 A-Maths programme](https://ancourage.academy/courses/academy/secondary/s4/a-maths) drills both procedures directly — [book a trial class (usually $18)](https://ancourage.academy/trial-class) for a diagnostic assessment.**

## What Do Polynomials Cover in A-Maths?

**In O-Level / SEC A-Maths, the polynomials strand covers multiplication and division of polynomials, the remainder and factor theorems, factorising polynomials, solving cubic equations, and the sum and difference of two cubes.** The [SEAB A-Maths syllabus (4049)](https://www.seab.gov.sg/gce-o-level/o-level-syllabuses-examined-for-school-candidates-2026/) defines exactly what is examinable, and from 2027 the same content carries over to the SEC G3 A-Maths syllabus (K341) unchanged.

## What Are the Remainder and Factor Theorems?

**The remainder theorem says that when a polynomial f(x) is divided by (x − a), the remainder is f(a); the factor theorem is the special case where the remainder is zero, so (x − a) is a factor exactly when f(a) = 0.**

-   **Remainder theorem:** the remainder on dividing f(x) by (x − a) is f(a). For a divisor (bx − a), the remainder is f(a/b).
-   **Factor theorem:** (x − a) is a factor of f(x) if and only if f(a) = 0.

These two results let you find unknown coefficients (by forming equations from given remainders or factors) without long division, and they are the starting point for solving cubics.

## How Do You Solve a Cubic Equation?

**To solve a cubic, use the factor theorem to find one root by trial, divide out that factor to get a quadratic, then solve the quadratic.**

1.  **Find a root by trial:** test small integer values (±1, ±2, factors of the constant term) until f(a) = 0. Then (x − a) is a factor.
2.  **Divide:** use long division or comparing coefficients to write f(x) = (x − a)(quadratic).
3.  **Solve the quadratic:** factorise or use the formula to find the remaining roots.

The sum and difference of two cubes are useful shortcuts that the syllabus names explicitly: a³ + b³ = (a + b)(a² − ab + b²) and a³ − b³ = (a − b)(a² + ab + b²). Recognising a cubic as one of these forms can skip the trial step entirely.

## What Are Partial Fractions?

**Partial fractions reverse the process of adding algebraic fractions — they split a single rational expression into a sum of simpler fractions, which is essential for integration and series work later.** The form of the decomposition depends entirely on the factors in the denominator, and the A-Maths syllabus tests three cases.

| Denominator type | Partial-fraction form |
| --- | --- |
| Distinct linear factors, e.g. (x + 1)(x − 2) | A/(x + 1) + B/(x − 2) |
| Repeated linear factor, e.g. (x − 3)² | A/(x − 3) + B/(x − 3)² |
| Non-factorisable quadratic factor, e.g. (x² + 1) | (Ax + B)/(x² + 1) |

One condition must be checked first: the fraction must be _proper_ (the numerator's degree lower than the denominator's). If it is improper, divide first, then decompose the proper remainder.

## How Do You Find the Unknown Constants?

**After writing the correct form, multiply through by the denominator and find the constants by substituting strategic values of x or by comparing coefficients.**

-   **Substitution (cover-up):** substitute the value of x that makes one factor zero to isolate a single constant quickly — the fastest method for distinct linear factors.
-   **Comparing coefficients:** expand and match the coefficients of each power of x — necessary for the quadratic-factor and repeated-factor cases.
-   **Mix both:** use substitution for the easy constants, then comparison for the rest.

## The Most Common Polynomials and Partial-Fraction Mistakes

**In our A-Maths classes at Ancourage Academy, a handful of recurring errors cause most avoidable mark loss in this topic.**

| Mistake | Why it happens | How to fix it |
| --- | --- | --- |
| Wrong remainder for (bx − a) | Substituting a instead of a/b | Set the divisor to zero: bx − a = 0 gives x = a/b, so the remainder is f(a/b) |
| Missing the repeated-factor term | Writing only A/(x − 3) for (x − 3)² | Include both A/(x − 3) and B/(x − 3)² |
| Decomposing an improper fraction directly | Not checking numerator vs denominator degree | Divide first if improper, then decompose the proper part |
| Linear numerator over a quadratic factor | Using A instead of (Ax + B) | A non-factorisable quadratic factor needs a linear numerator (Ax + B) |
| Stopping after one cubic root | Forgetting to divide and solve the quadratic | A cubic has up to three roots — always factor out and solve the quadratic |

## How Does This Topic Connect to the Rest of A-Maths?

**Partial fractions and polynomials feed directly into later A-Maths work and into JC.**

-   **Calculus:** beyond their algebra, partial fractions become a key integration tool at JC/H2 — splitting a fraction is often the only way to integrate it. See our [calculus deep-dive](https://ancourage.academy/articles/a-maths-calculus-differentiation-integration-singapore).
-   **Binomial theorem:** decomposed fractions can be expanded term by term, linking to the [binomial theorem](https://ancourage.academy/articles/a-maths-binomial-theorem-expansion-guide-singapore).
-   **Foundation for JC:** partial fractions are assumed knowledge in [H2 Mathematics](https://ancourage.academy/articles/h2-mathematics-jc-guide-singapore), where they appear in integration techniques.

## A Study Plan for Mastering Polynomials and Partial Fractions

**Work this topic in order: division and the theorems, then cubics, then partial fractions.**

1.  **Week 1 — division and theorems:** practise polynomial long division and the remainder and factor theorems, including the (bx − a) case.
2.  **Week 2 — cubics:** solve cubic equations by finding one root, dividing, and solving the quadratic; learn the sum and difference of cubes.
3.  **Week 3 — partial fractions:** drill the three denominator cases and both methods for finding the constants.
4.  **Week 4 — mixed practice:** tackle combined algebra questions, including improper fractions and partial-fraction decomposition, under timed conditions.

Ancourage Academy's [Sec 3](https://ancourage.academy/courses/academy/secondary/s3/a-maths) and [Sec 4 A-Maths](https://ancourage.academy/courses/academy/secondary/s4/a-maths) programmes work through this topic on exactly this progression in small groups of 3–6. If your child got stuck here, our [A-Maths survival guide](https://ancourage.academy/articles/a-maths-survival-guide-struggling-additional-maths-singapore) covers the wider recovery plan — book a [trial class (usually $18)](https://ancourage.academy/trial-class) for a diagnostic, or [WhatsApp us](https://api.whatsapp.com/send/?phone=6588498106&type=phone_number&app_absent=0) with any questions.

## Common Questions About A-Maths Polynomials and Partial Fractions

### What is the difference between the remainder and factor theorems?

The remainder theorem gives the remainder when f(x) is divided by (x − a): it equals f(a). The factor theorem is the special case where that remainder is zero — if f(a) = 0, then (x − a) is a factor of f(x). In short, the factor theorem is the remainder theorem applied to detect exact divisibility, and it is the standard way to start solving a cubic.

### How do you solve a cubic equation in A-Maths?

Use the factor theorem to find one root by testing small values such as ±1 and ±2 until f(a) = 0, so (x − a) is a factor. Divide f(x) by (x − a) using long division or by comparing coefficients to obtain a quadratic factor, then solve that quadratic by factorising or the formula. A cubic has up to three real roots, so do not stop after finding one.

### How do you choose the right partial-fraction form?

The form is decided by the denominator. Distinct linear factors each get a constant numerator, e.g. A/(x + 1) + B/(x − 2). A repeated linear factor needs two terms: A/(x − 3) + B/(x − 3)². A non-factorisable quadratic factor needs a linear numerator: (Ax + B)/(x² + 1). First make sure the fraction is proper; if not, divide before decomposing.

### When should you use the cover-up method versus comparing coefficients?

The cover-up (substitution) method is fastest for distinct linear factors: substitute the x-value that makes a factor zero to isolate each constant. Comparing coefficients is needed for repeated-factor and quadratic-factor cases, where substitution alone cannot find every constant. In practice, use substitution for the easy constants and comparison for the rest.

### Is this topic the same under SEC from 2027?

Yes. Polynomials and partial fractions move from the O-Level A-Maths syllabus (4049) to the SEC G3 A-Maths syllabus (K341) from 2027 with no change to the remainder and factor theorems, cubic-solving, or partial-fraction requirements. The "O-Level / SEC" dual reference reflects this transition.

Related: [A-Maths Quadratics](https://ancourage.academy/articles/a-maths-quadratic-functions-equations-inequalities-singapore) · [A-Maths Binomial Theorem](https://ancourage.academy/articles/a-maths-binomial-theorem-expansion-guide-singapore) · [Calculus in A-Maths](https://ancourage.academy/articles/a-maths-calculus-differentiation-integration-singapore) · [Surviving Additional Maths](https://ancourage.academy/articles/a-maths-survival-guide-struggling-additional-maths-singapore) · [E-Maths or A-Maths?](https://ancourage.academy/articles/e-maths-vs-a-maths-difference-singapore) · [a guide to A-Maths Indices, Surds, Logarithms & Exponentials](https://ancourage.academy/articles/a-maths-indices-surds-logarithms-exponentials-singapore)

## Related Courses

- [Sec 3 O-Level / SEC A-Maths](https://ancourage.academy/courses/academy/secondary/s3/a-maths) — Remainder and factor theorems and cubic equations in small groups of 3–6
- [Sec 4 O-Level / SEC A-Maths](https://ancourage.academy/courses/academy/secondary/s4/a-maths) — Partial fractions and exam preparation
- [Secondary A-Maths Programme](https://ancourage.academy/courses/academy/secondary/a-maths) — All A-Maths courses by level at Bishan and Woodlands
- [Trial Class (Usually $18)](https://ancourage.academy/trial-class) — Diagnostic assessment of your child’s polynomials foundations

## Sources

- [O-Level Additional Mathematics Syllabus 4049 (seab.gov.sg)](https://www.seab.gov.sg/gce-o-level/o-level-syllabuses-examined-for-school-candidates-2026/) — Singapore Examinations and Assessment Board
- [SEC G3 Additional Mathematics Syllabus K341 (seab.gov.sg)](https://www.seab.gov.sg/secondary-education-certificate-sec/g3-syllabuses-for-school-candidates-2027/) — Singapore Examinations and Assessment Board
- [Secondary Mathematics Syllabuses (moe.gov.sg)](https://www.moe.gov.sg/secondary/courses) — Ministry of Education, Singapore