MathematicsChapter 5: Polynomials

Remainder Theorem

When p(x) is divided by x-c, the remainder is p(c).

Learning objectives

State the main ideas and notation used in remainder theorem.
Apply the rules of remainder theorem in a clear, step-by-step solution.
Check signs, conditions, and the final answer for mathematical accuracy.

Definition

When p(x) is divided by x-c, the remainder is p(c). You do not need to complete the long division to know that leftover number. Meaning: plug in the root of the divisor. If you divide by ax-b, the remainder is p(ba) (still a constant). Proof idea: p(x)=(x-c)q(x)+R, put x=c, get p(c)=R.

Where / symbols

REMAINDER THEOREM: When p(x) is divided by x-c, the remainder is p(c). You do not need to complete the long division to know that leftover number. Meaning: plug in the root of the divisor. If you divide by ax-b, the remainder is p(b/a) (still a constant). Proof idea: p(x)=(x-c)q(x)+R, put x=c, get p(c)=R.
1. WHY: Identity p(x)=(x-c)q(x)+R. At x=c the first piece dies. R is constant because the divisor is degree 1. That is the whole theorem.
2. FIND REMAINDERS: p(x)=x^3-2x+5 divided by x-1: remainder p(1)=4. By x+2 i.e. x-(-2): p(-2). By 2x-1: remainder p(1/2).

Overview

Why it works, then use it:

Why

Identity p(x)=(x-c)q(x)+R. At x=c the first piece dies. R is constant because the divisor is degree 1. That is the whole theorem.

Examples No long division needed · c is the 'plug-in' · R a number

Find Remainders

p(x)=x³-2x+5 divided by x-1: remainder p(1)=4. By x+2 i.e. x-(-2): p(-2). By 2x-1: remainder p(12).

Examples x-1 -> p(1) · x+3 -> p(-3) · 2x-1 -> p(1/2)

WHAT — REMAINDER THEOREM

RULE — 1. WHY

Identity p(x)=(x-c)q(x)+R. At x=c the first piece dies. R is constant because the divisor is degree 1. That is the whole theorem.

METHOD — 2. FIND REMAINDERS

p(x)=x³-2x+5 divided by x-1: remainder p(1)=4. By x+2 i.e. x-(-2): p(-2). By 2x-1: remainder p(12).

WORKED EXAMPLE

Worked: remainder of x^3 - 4x + 1 divided by x-2. p(2) = 8 - 8 + 1 = 1 remainder 1

How to answer a remainder theorem question

Identify the given quantities, notation, and required result.
Choose the matching rule and write it before substitution.
Work one algebraic step at a time, keeping brackets and signs clear.
Check restrictions, simplify the result, and verify it where possible.

Worked example

remainder of x^3 - 4x + 1 divided by x-2.
p(2) = 8 - 8 + 1
= 1
remainder 1

In simple words

Dividing by x+c is dividing by x-(-c). Plug in -c, not +c.

Key points

divide by x-c: remainder p(c)
just substitute
p(c)=R
match x-c

Exam preparation

Exam Focus

Quick revision

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Review the key facts, formulas and checks before you move on.