Polynomial Long Division: Step-by-Step Method & Worked Example

Q: What is polynomial long division?

Polynomial long division is an algorithm for dividing one polynomial P(x) by another S(x), producing a quotient Q(x) and a remainder R(x) with deg(R) < deg(S), satisfying P(x) = Q(x)·S(x) + R(x). It works by repeatedly dividing the leading term of the current remainder by the leading term of the divisor.

Q: How do you know when to stop polynomial long division?

Stop when the degree of the current remainder is strictly less than the degree of the divisor. At that point, the remainder is R(x) and the accumulated terms form the quotient Q(x). Continuing further would produce terms with negative exponents, which aren't polynomials.

Q: What are the degrees of the quotient and remainder?

If P has degree m and S has degree n (with m ≥ n), then the quotient Q has degree m − n, and the remainder R has degree strictly less than n (or R = 0). For example, dividing a degree-3 polynomial by a degree-2 polynomial gives a degree-1 quotient and a remainder of degree at most 1.

Q: Why do you need zero placeholders in long division?

If the dividend is missing some powers of x (e.g., x⁴ + 1 has no x³, x², or x terms), you must write it as x⁴ + 0x³ + 0x² + 0x + 1 in descending order. This keeps columns aligned so that like terms line up for subtraction. Without placeholders, the algorithm's bookkeeping breaks down.

What Is Polynomial Division?

Dividing a polynomial \( P(x) \) by a polynomial \( S(x) \) gives the identity:

\[ \dfrac{P(x)}{S(x)} = Q(x) + \dfrac{R(x)}{S(x)} \]

Multiplying both sides by \( S(x) \):

\[ P(x) = Q(x)\,S(x) + R(x) \]

Dividend \( P(x) \) — the polynomial being divided.
Divisor \( S(x) \) — the polynomial you divide by.
Quotient \( Q(x) \) — the result of the division.
Remainder \( R(x) \) — what is left over, with \( \deg(R) < \deg(S) \).

Degree bounds: if \( \deg(P) = m \) and \( \deg(S) = n \) with \( m \ge n \), then \( \deg(Q) = m - n \) and \( \deg(R) < n \). These bounds are what make long division terminate.

Preparation for Long Division

Before starting, write both the dividend and the divisor in descending powers of \( x \). Use \( 0 \) as a coefficient placeholder for any missing powers, so that like terms align during the subtraction steps. For example:

\[ x^4 + 1 \quad \text{becomes} \quad x^4 + 0x^3 + 0x^2 + 0x + 1 \]

Without placeholders, the bookkeeping breaks down when subtracting.

Worked Example

Divide \( 2x^3 + 3x^2 - x + 16 \) by \( x^2 + 2x - 3 \).

Six-step solution

The dividend \( 2x^3 + 3x^2 - x + 16 \) has degree 3; the divisor \( x^2 + 2x - 3 \) has degree 2. So the quotient will have degree \( 3 - 2 = 1 \), and the remainder will have degree at most 1.

Step 1: divide the leading term 2x³ by x² to get the first quotient term 2x — **Step 1 — Divide leading terms.** \( 2x^3 \div x^2 = 2x \). This is the first term of the quotient.

Step 2: multiply the divisor by 2x, giving 2x³ + 4x² − 6x, aligned under the dividend — **Step 2 — Multiply.** \( 2x \cdot (x^2 + 2x - 3) = 2x^3 + 4x^2 - 6x \). Align under the dividend.

Step 3: subtract the product from the dividend, giving −x² + 5x + 16 — **Step 3 — Subtract.** Be careful to distribute the negative to every term: \( (2x^3 + 3x^2 - x + 16) - (2x^3 + 4x^2 - 6x) = -x^2 + 5x + 16 \).

Step 4: bring down the next terms and divide −x² by x² to get −1 — **Step 4 — Divide again.** Bring down any remaining terms, then divide the new leading term: \( -x^2 \div x^2 = -1 \). This is the next quotient term.

Step 5: multiply the divisor by −1, giving −x² − 2x + 3 — **Step 5 — Multiply.** \( -1 \cdot (x^2 + 2x - 3) = -x^2 - 2x + 3 \).

Step 6: subtract to find the final remainder 7x + 13 — **Step 6 — Subtract to find the remainder.** \( (-x^2 + 5x + 16) - (-x^2 - 2x + 3) = 7x + 13 \).

Stop. The remainder \( 7x + 13 \) has degree 1, which is less than \( \deg(S) = 2 \). Long division is complete. (Continuing would produce terms with negative exponents — not polynomials.)

Final answer:

\[ \dfrac{2x^3 + 3x^2 - x + 16}{x^2 + 2x - 3} = 2x - 1 + \dfrac{7x + 13}{x^2 + 2x - 3} \]

In multiplication form:

\[ 2x^3 + 3x^2 - x + 16 = (2x - 1)(x^2 + 2x - 3) + (7x + 13) \]

Check: \( (2x - 1)(x^2 + 2x - 3) + (7x + 13) \) expands to \( 2x^3 + 3x^2 - x + 16 \) ✓

Practice Exercises

Use the six-step algorithm on each problem, then expand the answer to check your work.

Exercise 1 — Divide \( x^3 - 1 \) by \( x - 1 \)

Answer

Write the dividend with placeholders: \( x^3 + 0x^2 + 0x - 1 \).

\[ Q(x) = x^2 + x + 1, \qquad R(x) = 0 \]

Equivalently: \( x^3 - 1 = (x - 1)(x^2 + x + 1) \). This is the standard difference-of-cubes factorization.

Exercise 2 — Divide \( x^4 + 1 \) by \( x - 1 \)

(Demonstrates why placeholders matter!)

Answer

Write the dividend as \( x^4 + 0x^3 + 0x^2 + 0x + 1 \).

\[ Q(x) = x^3 + x^2 + x + 1, \qquad R(x) = 2 \]

Equivalently: \( x^4 + 1 = (x - 1)(x^3 + x^2 + x + 1) + 2 \).

(By the Remainder Theorem: \( P(1) = 1^4 + 1 = 2 = R \) ✓)

Exercise 3 — Divide \( 2x^3 + 3x^2 - x + 16 \) by \( x^2 + 2x - 3 \)

(Same as the worked example — try it yourself before revealing.)

Answer

\[ Q(x) = 2x - 1, \qquad R(x) = 7x + 13 \]

Frequently Asked Questions

What is polynomial long division?

An algorithm for dividing \(P(x)\) by \(S(x)\), producing a quotient \(Q(x)\) and remainder \(R(x)\) with \(\deg(R) < \deg(S)\), such that \(P(x) = Q(x) \cdot S(x) + R(x)\). It works by repeatedly dividing the leading term of the current remainder by the leading term of the divisor.

How do you know when to stop polynomial long division?

Stop when the degree of the current remainder is strictly less than the degree of the divisor. At that point, the remainder is \(R(x)\) and the accumulated terms form \(Q(x)\). Continuing would produce negative-exponent terms, which aren't polynomials.

What are the degrees of the quotient and remainder?

If \(P\) has degree \(m\) and \(S\) has degree \(n\) (with \(m \ge n\)), then \(Q\) has degree \(m - n\), and \(R\) has degree at most \(n - 1\) (or \(R = 0\)).

Why do you need zero placeholders in long division?

If the dividend is missing some powers of \(x\), write them as \(0x^k\) terms. For example, \(x^4 + 1\) becomes \(x^4 + 0x^3 + 0x^2 + 0x + 1\). This keeps columns aligned for subtraction; without it, the bookkeeping breaks.

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Last updated: October 17, 2026