Polynomial Division (P3 Algebra 3.1) — Mathematics

Introduction

Polynomial division is the algebraic process of dividing one polynomial by another of lower degree, producing a quotient and a remainder — exactly analogous to integer long division. In the 9709 examination, you will divide polynomials of degree at most 4 by either a linear polynomial (degree 1) or a quadratic polynomial (degree 2). The result always takes the form:

\frac{P(x)}{D(x)} = Q(x) + \frac{R(x)}{D(x)}

where $P(x)$ is the dividend, $D(x)$ is the divisor, $Q(x)$ is the quotient, and $R(x)$ is the remainder. This identity underpins partial fractions, the Factor Theorem, and the Remainder Theorem — all heavily examined topics in P3.

Core Concept

Given a polynomial $P(x)$ of degree $n$ and a divisor $D(x)$ of degree $d$ (where $d \leq n$ ), polynomial long division produces a unique quotient $Q(x)$ of degree $n - d$ and a remainder $R(x)$ of degree strictly less than $d$ .

The key identity is:

P(x) = D(x) \cdot Q(x) + R(x)

This must hold for all values of $x$ .

Two standard methods are used in A-Level examinations:

Polynomial long division — the algorithmic column method, reliable for all cases.
Inspection / equating coefficients — writing $P(x) = D(x) \cdot Q(x) + R(x)$ with unknown coefficients and solving; efficient when the quotient structure is predictable.

Key cases covered by the syllabus:

Dividend degree	Divisor degree	Quotient degree	Remainder degree
2	1	1	0 (constant)
3	1	2	0 (constant)
4	1	3	0 (constant)
3	2	1	$\leq 1$ (linear or zero)
4	2	2	$\leq 1$ (linear or zero)

When the remainder is zero, the divisor is a factor of the dividend.

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Introduction

Core Concept

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