Notebook

00:03

12 Apr 2026

The Polynomial Division Algorithm

Welcome! Today we're exploring The Polynomial Division Algorithm — a powerful technique that extends something you already know into the world of polynomials.

You learned long division with numbers in primary school. Polynomial division follows the exact same idea, with one twist:

Instead of comparing sizes, you compare degrees.

Understanding the division algorithm statement is essential. It tells you:

When to stop dividing
What to expect from the quotient and remainder

Pay special attention to the degree condition on the remainder — this is the key detail that makes polynomial division work cleanly.

What you'll learn	Why it matters
Division algorithm statement	The foundation for all polynomial division
Degree condition on remainder	Tells you when you're done

1. The division algorithm statement

Let's talk about polynomial division

You already know long division with numbers from primary school. The polynomial division algorithm is the algebraic version of that same idea.

The statement has two parts:

The equation
The degree condition on the remainder

📋 Given Info

Think about integer division:

$17 = 3 \times 5 + 2$

Here, the remainder ( $2$ ) is less than the divisor ( $5$ ).

For polynomials, the same structure holds — but instead of comparing values, we compare degreeskey!.

✍️ Question

Your turn ✍️

State the division algorithm for polynomials.

What condition must the remainder satisfy?

The division algorithm for polynomials:

f(x) = q(x) \cdot g(x) + r(x)

(Your verification tool to check your answer)

where $r (x) = 0$ or $\deg(r) < \deg(g)$ Stop when remainder degree is smaller than divisor.

In words: Dividend = Quotient × Divisor + Remainder, and the remainder has degree STRICTLY LESSkey!Check this to know if you're done dividing than the divisor.

Just like in integer division where $17 = 3 \times 5 + 2$ (remainder 2 < divisor 5), polynomial division follows the same structure — but we compare degrees instead of values.

✍️ MCQ

Choose one

If you're dividing by a polynomial of degree

3

, what's the maximum possible degree of the remainder?

✓

Examples of the stopping condition:

Dividing by degree 1 (linear): stop when remainder is degree 0 (a constant).
Dividing by degree 2 (quadratic): stop when remainder is degree 0 or 1.
Dividing by degree 3: stop when remainder is degree 0, 1, or 2.

✍️ MCQ

Choose one

You're dividing a polynomial by a quadratic divisor (degree 2). Which of these could be a valid remainder?

✓

The key word is 'degree'Everything in polynomial division revolves around this — for polynomials, we compare degrees, not valuesKey difference from integer division.

2. Predicting quotient and remainder degrees

Before we execute a polynomial division, there's a useful skill: predicting the degrees of the quotient and remainder before you even start dividing.

These predictions act as sanity checks — if your final answer doesn't match the predicted degrees, you know something went wrong!

📋 Given Info

Here are the two key rules:

Quotient degree = $\deg (dividend) - \deg (divisor)$
Remainder degree $<$ $\deg (divisor)$

These are your sanity checks — if your final answer doesn't follow these rules, you made a mistake somewhere.

✍️ Question

Let's test this 🧪

Suppose you're dividing a degree-4 polynomial by a degree-2 polynomial.

(a) What degree will the quotient have?

(b) What is the maximum possible degree of the remainder?

Predicting degrees before dividing:

Quotient degree = degree(f) − degree(g)Use this to verify your answer is correct

For degree 4 ÷ degree 2: quotient has degree $4 - 2 = 2$ Calculate expected degree before starting.

So before we even start the long division, we know the quotient will be a degree-2 polynomial — a quadratic.Wrong degree means calculation error

✍️ MCQ

Choose one

When dividing by a degree-2 polynomial, what is the maximum possible degree of the remainder?

✓

Remainder degree: must be STRICTLY LESSNot less than or equal, but strictly less than degree(g).

For a degree-2 divisor: remainder degree < 2, so at most degree 1Linear, constant, or zero — nothing higher.

This means the remainder can be:

A linear polynomial (degree 1), like $3 x + 5$
A constant (degree 0), like $7$
Or simply $0$

✍️ MCQ

Choose one

If you divide a degree-5 polynomial by a degree-3 polynomial, what is the maximum possible degree of the remainder?

✓

The remainder could be:

A constant like $5$ (degree 0). Degree $0 < 2$ . ✅ Valid.

A linear expression like $2 x + 3$ (degree 1). Degree $1 < 2$ Maximum remainder degree is one less than divisor. ✅ Valid.

A quadraticIf remainder has same degree, keep dividing like $x^{2} + 1$ (degree 2). Degree $2$ is NOT less than $2$ Remainder degree must drop below divisor degree. ❌ Invalidwrong!Keep dividing until remainder degree is lower — you have not finished dividing.

✍️ MCQ

Choose one

You're dividing a polynomial by

x^2 + 1

. Which of these could be a valid remainder?

✓

💡 Pro tip: These predictions are sanity checksThese checks catch your mistakes automatically. If your quotient has the wrong degreeFirst question to ask yourself after dividing, or your remainder has degree ≥ divisor degreeSecond question to verify your work, you made an error somewhere — go back and check your workIf either check fails, find your mistake!