Polynomial Long Division — Mathematics Knowledge System

Overview

Polynomial long division divides one polynomial by another using the same step-by-step process as long division of numbers: divide, multiply, subtract, bring down, and repeat. It expresses any dividend f(x) as f(x) = d(x)·q(x) + r(x), where the remainder r(x) has degree strictly less than the divisor d(x) — the key tool for factoring higher-degree polynomials and finding rational roots.

Intuition

Just as 17 ÷ 5 gives quotient 3 and remainder 2 (17 = 5·3 + 2), dividing polynomials produces a quotient and a remainder, except now the 'size' being compared is degree rather than magnitude. You repeatedly cancel the highest-degree term of what's left using the highest-degree term of the divisor, until what remains has smaller degree than the divisor — at which point you can't divide any further.

Formal Definition

Definition

The Division Algorithm for polynomials: for any polynomials f(x) and d(x)≠0, there exist unique polynomials q(x) (quotient) and r(x) (remainder) such that:

f(x) = d(x)\,q(x) + r(x), \quad \deg(r) < \deg(d) \text{ or } r(x)=0

Division Algorithm

Properties

Uniqueness

q(x) \text{ and } r(x) \text{ are uniquely determined by } f(x) \text{ and } d(x)

Remainder Theorem (linear divisor)

\text{Dividing } f(x) \text{ by } (x-c) \text{ gives remainder } r = f(c)

Exact division

d(x) \text{ divides } f(x) \text{ evenly} \iff r(x) = 0

Applications

Polynomial division underlies error-correcting codes (e.g. CRC checksums, Reed–Solomon codes), where messages are divided by generator polynomials.

Worked Examples

Divide leading terms: x³÷x=x². Multiply x²(x-3)=x³-3x², subtract from f(x).
$(x^3-2x^2) - (x^3-3x^2) = x^2 \Rightarrow \text{bring down: } x^2-5x$
Divide leading terms: x²÷x=x. Multiply x(x-3)=x²-3x, subtract.
$(x^2-5x)-(x^2-3x) = -2x \Rightarrow \text{bring down: } -2x+6$
Divide leading terms: -2x÷x=-2. Multiply -2(x-3)=-2x+6, subtract to get remainder 0.
$(-2x+6)-(-2x+6) = 0$

Answer: Quotient: x² + x - 2, remainder 0 (so x-3 divides evenly)

Practice Problems

Difficulty 4/10

Divide f(x) = 2x³ + 3x² - x + 5 by d(x) = x + 2, and state the quotient and remainder.

Difficulty 3/10

Use the Remainder Theorem to find the remainder when f(x)=x⁴-3x+1 is divided by (x-2), without doing full long division.

Common Mistakes

Common Mistake

Forgetting to include placeholder terms with coefficient 0 for missing degrees (e.g. dividing x³+1 without accounting for the missing x² and x terms).

Always rewrite the dividend with every degree present, using 0 as the coefficient for missing terms: x³+1 = x³+0x²+0x+1. Skipping this causes terms to align incorrectly during subtraction.

Common Mistake

Adding instead of subtracting when eliminating the leading term at each step.

After multiplying the divisor by the current quotient term, you must SUBTRACT that product from the current remainder — sign errors here are the most common source of mistakes in polynomial long division.

Summary

Polynomial long division follows divide–multiply–subtract–bring down, just like numeric long division.
The Division Algorithm: f(x) = d(x)q(x) + r(x), with deg(r) < deg(d).
The Remainder Theorem gives a shortcut: dividing by (x-c) leaves remainder f(c), no full division needed.
Always fill in zero-coefficient placeholders for missing-degree terms before dividing.

References

BookSullivan, M. Algebra and Trigonometry, 10th ed. Ch. 5 — Polynomial and Rational Functions.
WebsiteWikipedia — Polynomial long division