# Factors of Polynomials

> CIE IGCSE Additional Mathematics · CIE IGCSE Add Maths (0606)
> Source: https://www.owlsprep.com/study/cie-0606-u3-overview/
> Weight: ~5-7% of Paper 1 and Paper 2 structured questions

This unit introduces core algebraic tools for working with polynomials up to degree 3, enabling you to factorise expressions and solve higher-degree equations efficiently, a foundational skill for later calculus and coordinate geometry topics.

**Prerequisites:** Expanding and simplifying algebraic expressions; Solving linear and quadratic equations

## Learning objectives

- Apply the Remainder Theorem to calculate the remainder when a polynomial is divided by a linear factor
- Use the Factor Theorem to identify linear factors of polynomials up to degree 3
- Fully factorise cubic polynomials and solve cubic equations using algebraic methods

## Unit at a glance

Polynomials are expressions of the form $a_nx^n + a_{n-1}x^{n-1} + ... + a_0$, 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](https://www.owlsprep.com/study/cie-0606-u3-factor-remainder-theorems-and-cubic/) — Learn the Remainder and Factor Theorems, factorise cubic polynomials, and solve cubic equations using algebraic methods.

## Common pitfalls

- **Wrong:** Forgetting the sign when applying the Remainder Theorem for divisor $(x + a)$
  - Why it fails: The remainder is $f(-a)$, not $f(a)$, so sign errors lead to incorrect remainder calculations
  - Correct: Always substitute $x = -a$ when dividing by $(x + a)$, and $x = a$ when dividing by $(x - a)$
- **Wrong:** Stopping factorisation of a cubic after finding one factor, leaving a quadratic un-factorised if possible
  - Why it fails: Exam questions frequently require fully factorised form or all roots of the cubic
  - Correct: After extracting one linear factor, factorise the resulting quadratic fully if it has real, integer roots
- **Wrong:** Dividing a polynomial by a non-linear factor and attempting to apply the Factor/Remainder Theorems
  - Why it fails: The theorems only apply to linear divisors of the form $(ax + b)$
  - Correct: Use polynomial long division or equate coefficients for non-linear divisors, and restrict Factor/Remainder Theorem use to linear divisors

## Cheatsheet

| Name | Formula/Rule | Use Case |
| --- | --- | --- |
| Remainder Theorem | When $f(x)$ is divided by $(ax - b)$, remainder = $f\left(\frac{b}{a}\right)$ | Find the remainder of polynomial division without full long division |
| Factor Theorem | $(ax - b)$ is a factor of $f(x)$ if and only if $f\left(\frac{b}{a}\right) = 0$ | Identify linear factors of polynomials up to degree 3 |
| Cubic Factorisation | $ax^3 + bx^2 + cx + d = (px + q)(rx^2 + sx + t)$ | Break down cubics into products of linear and quadratic factors |
| Cubic Equation Solution | If $(ax - b)(cx^2 + dx + e) = 0$, roots are $x = \frac{b}{a}$ and roots of $cx^2 + dx + e = 0$ | 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.

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