Unit Overview

Factors of Polynomials

CIE IGCSE Additional Mathematics· 5 min read 📊 ~5-7% of Paper 1 and Paper 2 structured questions

1. Unit at a glance

Polynomials are expressions of the form $a_{n} x^{n} + a_{n - 1} x^{n - 1} + ... + a_{0}$ , and working with their factors is a key skill for solving complex algebraic problems across the syllabus. This unit builds directly on your existing knowledge of quadratic factorisation to extend your skill set to cubic expressions.

All content for this unit is contained in a single sub-topic that combines teaching of the two core theorems, practical factorisation techniques, and equation solving, with exam-aligned examples to reinforce understanding.

The only sub-topic in this unit covers all required content for polynomial factors:

Factor & Remainder Theorems and Cubic Equations

Learn the Remainder and Factor Theorems, factorise cubic polynomials, and solve cubic equations using algebraic methods.

★★⏱ 12 min

2. Common Pitfalls

Wrong move:

Forgetting the sign when applying the Remainder Theorem for divisor $(x + a)$

Why:

The remainder is $f (- a)$ , not $f (a)$ , so sign errors lead to incorrect remainder calculations

Correct move:

Always substitute $x = - a$ when dividing by $(x + a)$ , and $x = a$ when dividing by $(x - a)$

Wrong move:

Stopping factorisation of a cubic after finding one factor, leaving a quadratic un-factorised if possible

Why:

Exam questions frequently require fully factorised form or all roots of the cubic

Correct move:

After extracting one linear factor, factorise the resulting quadratic fully if it has real, integer roots

Wrong move:

Dividing a polynomial by a non-linear factor and attempting to apply the Factor/Remainder Theorems

Why:

The theorems only apply to linear divisors of the form $(a x + b)$

Correct move:

Use polynomial long division or equate coefficients for non-linear divisors, and restrict Factor/Remainder Theorem use to linear divisors

3. Quick Reference Cheatsheet

Name	Formula/Rule	Use Case
Remainder Theorem	When $f (x)$ is divided by $(a x - b)$ , remainder = $f (\frac{b}{a})$	Find the remainder of polynomial division without full long division
Factor Theorem	$(a x - b)$ is a factor of $f (x)$ if and only if $f (\frac{b}{a}) = 0$	Identify linear factors of polynomials up to degree 3
Cubic Factorisation	$a x^{3} + b x^{2} + c x + d = (p x + q) (r x^{2} + s x + t)$	Break down cubics into products of linear and quadratic factors
Cubic Equation Solution	If $(a x - b) (c x^{2} + d x + e) = 0$ , roots are $x = \frac{b}{a}$ and roots of $c x^{2} + d x + e = 0$	Find all real or integer roots of cubic equations

What's Next

You are ready to begin the only sub-topic in this unit, which covers all required content for factors of polynomials for the CIE IGCSE Additional Mathematics syllabus. Mastering this content will prepare you for upcoming units on coordinate geometry and calculus, where polynomial factorisation is used regularly to solve a wide range of problems.