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, whose secondary A-Maths tuition teaches these procedures step by step in small groups of 3–6 at Bishan and Woodlands.
This is a single-topic deep-dive — a sibling to our A-Maths quadratics and binomial theorem guides. If you are still deciding whether to take A-Maths, read E-Maths vs A-Maths first.
If cubic equations or partial fractions are a gap, Ancourage Academy's Sec 4 A-Maths programme drills both procedures directly — book a trial class (usually $18) 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) 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.
- 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.
- Divide: use long division or comparing coefficients to write f(x) = (x − a)(quadratic).
- 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.
- Binomial theorem: decomposed fractions can be expanded term by term, linking to the binomial theorem.
- Foundation for JC: partial fractions are assumed knowledge in H2 Mathematics, 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.
- Week 1 — division and theorems: practise polynomial long division and the remainder and factor theorems, including the (bx − a) case.
- Week 2 — cubics: solve cubic equations by finding one root, dividing, and solving the quadratic; learn the sum and difference of cubes.
- Week 3 — partial fractions: drill the three denominator cases and both methods for finding the constants.
- Week 4 — mixed practice: tackle combined algebra questions, including improper fractions and partial-fraction decomposition, under timed conditions.
Ancourage Academy's Sec 3 and Sec 4 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 covers the wider recovery plan — book a trial class (usually $18) for a diagnostic, or WhatsApp us 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 · A-Maths Binomial Theorem · Calculus in A-Maths · Surviving Additional Maths · E-Maths or A-Maths? · a guide to A-Maths Indices, Surds, Logarithms & Exponentials