Factors of Polynomials

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Cambridge (CIE) IGCSE · Additional Maths

Factors of Polynomials

The Big Idea: Any polynomial can be broken down (factorised) into a chain of simpler pieces multiplied together — and the fastest way to find those pieces is to smartly guess a root, confirm it with the factor theorem, then peel it off using polynomial division. Repeat until nothing more will factorise.

Summary — What This Chapter Covers

Operations with polynomials — what counts as a polynomial, and how to expand brackets with more than two terms/factors.
Polynomial division — dividing one polynomial by another (linear or quadratic) using the "bus stop" method, producing a quotient and a remainder.
The Factor Theorem — a fast test for whether (x − p) is a factor of a polynomial, without doing full division.
The Remainder Theorem — a way to find the remainder of a division instantly, and to solve for unknown coefficients.
Solving cubic equations — combining the factor theorem + division + quadratic formula to fully factorise and solve cubics.

1. Operations with Polynomials

What actually IS a polynomial?

A polynomial is an algebraic expression made of a finite number of terms, where every power (index) on the variable is a non-negative integer (0, 1, 2, 3, ...). That's the whole test — no fractional powers, no negative powers.

Think of it like a guest list rule at a party: only whole, non-negative numbers get an invite as exponents. Square roots (fractional powers) and "over x" terms (negative powers) are turned away at the door.

✅ These ARE polynomials 3x + 5 2x²y − 4y + 6 8 (yes — a plain number is a "constant" polynomial!)

❌ These are NOT polynomials √x (= x^½, a fractional index) 5x⁻³ (negative index) 3/x (= 3x⁻¹, negative index)

Why This Matters

Recognising polynomials matters because every rule in this chapter — division, factor theorem, remainder theorem — only applies to polynomials. If you're handed something with a √x or 1/x in it, these tools don't directly apply.

Expanding brackets with more than 2 factors

You already know FOIL for two brackets like (x + 2)(x + 3). But FOIL is really just a shortcut for a much more general rule: every term in the first bracket multiplies every term in every other bracket.

When you have three or more brackets, the trick is to only ever work with a pair at a time. Expand two of them fully first, simplify, then multiply that result by the next bracket.

General Rule (a + b)(x + y + z) = a(x + y + z) + b(x + y + z) = ax + ay + az + bx + by + bz In plain words: distribute EVERY term in bracket 1 across EVERY term in bracket 2 — nothing skipped.

Worked Example — Three Brackets

Expand (x − 4)(x + 3)(x² + x − 1)

Step 1 — pick a pair: expand (x − 4)(x + 3) first → x² − x − 12

Step 2 — multiply that result by the third bracket:
(x² − x − 12)(x² + x − 1)
= x²(x² + x − 1) − x(x² + x − 1) − 12(x² + x − 1)
= x⁴ + x³ − x² − x³ − x² + x − 12x² − 12x + 12

Step 3 — collect like terms:
= x⁴ − 14x² − 11x + 12

Common Mistake

two brackets first

Practice Question 1

Expand and simplify: 4x(x − 3y)(3x + 2y − 5)

Practice Question 2

Is 6x⁻² + 5x + 1 a polynomial? Explain why or why not.

2. Polynomial Division

The Factor Theorem

(x − p)

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Also in the full note

3. Factor & Remainder Theorem
4. Solving Cubic Equations
What to Memorise
Concepts Checklist
Exam Tips
The method, step by step
Dividing by a quadratic divisor

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Factors of Polynomials