MathematicsChapter 5: Polynomials

Synthetic Division (school)

Synthetic division is a short table for dividing by x-c.

Learning objectives

State the main ideas and notation used in synthetic division (school).
Apply the rules of synthetic division (school) in a clear, step-by-step solution.
Check signs, conditions, and the final answer for mathematical accuracy.

Definition

Synthetic division is a short table for dividing by x-c. Write c on the left, coefficients in a row (include 0 for missing powers). Bring down, multiply by c, add, repeat. Last number is the remainder (and p(c)). The other numbers are coefficients of the quotient, degree one less. Meaning: same arithmetic as long division, less writing.

Formula

c=2, coeffs 1 0 -3 -2

Where / symbols

SYNTHETIC DIVISION (SCHOOL): Synthetic division is a short table for dividing by x-c. Write c on the left, coefficients in a row (include 0 for missing powers). Bring down, multiply by c, add, repeat. Last number is the remainder (and p(c)). The other numbers are coefficients of the quotient, degree one less. Meaning: same arithmetic as long division, less writing.
1. TABLE: Missing x^2 in a cubic: write 0. Signs matter. If dividing by x+2, c=-2. Bring down the first coefficient unchanged.
2. QUOTIENT: If p was degree 3, quotient is degree 2 with those first leftover numbers. Then continue factoring the quadratic. Check by expanding back if marks allow.

Relationship

last output = remainder = p(c)

Overview

Set the row, then read the answer:

Table

Missing x^2 in a cubic: write 0. Signs matter. If dividing by x+2, c=-2. Bring down the first coefficient unchanged.

Examples x-2 -> c=2 · x+1 -> c=-1 · Last = remainder

Quotient

If p was degree 3, quotient is degree 2 with those first leftover numbers. Then continue factoring the quadratic. Check by expanding back if marks allow.

Examples Numbers are coeffs · Remainder right end · p(c) check

WHAT — SYNTHETIC DIVISION (SCHOOL)

RULE — 1. TABLE

Missing x^2 in a cubic: write 0. Signs matter. If dividing by x+2, c=-2. Bring down the first coefficient unchanged.

METHOD — 2. QUOTIENT

If p was degree 3, quotient is degree 2 with those first leftover numbers. Then continue factoring the quadratic. Check by expanding back if marks allow.

WORKED EXAMPLE

Worked: divide x^3 + 0 x^2 - 3x - 2 by x-2 (from earlier). c=2, coeffs 1 0 -3 -2 bring 1; 2 add -> 2; 2 add -> 1; *2 add -> 0 quotient x^2 + 2x + 1, remainder 0

How to answer a synthetic division (school) 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

divide x^3 + 0 x^2 - 3x - 2 by x-2 (from earlier).
c=2, coeffs 1 0 -3 -2
bring 1; 2 add -> 2; 2 add -> 1; *2 add -> 0
quotient x^2 + 2x + 1, remainder 0

In simple words

For x+3, the left number is -3, not +3.

Key points

c on the left, coeffs in a row
last output = remainder = p(c)
include zeros
one degree less

Exam preparation

Exam Focus

Quick revision

Revision Zone

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