Polynomial Functions

Polynomial Long Division

Dividing a polynomial by a binomial using long division.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Multiplicity of Zeros

The Same Long Division, New Objects

You already know how to divide 527 by 4 using long division — matching place value at each step, subtracting, and bringing down the next digit. Before reading on, think about whether that exact same process could apply to dividing a polynomial like x³ + 2x² − 5x + 6 by x − 2, matching degree the way numeric division matches place value.

It can, term for term. Numeric long division asks, at each step, 'how many times does the divisor go into the current leading digits,' subtracts, and continues with what's left. Polynomial long division asks the same question about leading terms: divide the dividend's leading term by the divisor's leading term, multiply that result back through the whole divisor, subtract, and repeat with what remains — degree playing the exact role place value played before.

Worked Example — Dividing a Cubic by a Linear Binomial

Divide x³ + 2x² − 5x + 6 by x − 2. Divide leading terms: x³ ÷ x = x². Multiply x²(x − 2) = x³ − 2x², subtract from the dividend: (x³ + 2x²) − (x³ − 2x²) = 4x², bringing down the rest: 4x² − 5x + 6. Repeat: 4x² ÷ x = 4x. Multiply 4x(x − 2) = 4x² − 8x, subtract: 3x + 6. Repeat: 3x ÷ x = 3. Multiply 3(x − 2) = 3x − 6, subtract: 12. The quotient is x² + 4x + 3, with a remainder of 12 — written as a full expression, x³ + 2x² − 5x + 6 = (x − 2)(x² + 4x + 3) + 12.

Worked Example — Dividing with a Nonzero Remainder

Divide 2x² + 3x − 1 by x + 2. Following the same process gives a quotient of 2x − 1 with a remainder of 1. Written as a full expression: 2x² + 3x − 1 = (x + 2)(2x − 1) + 1, matching the same 'quotient times divisor plus remainder' structure as numeric division.

Tip

Include a placeholder term with coefficient 0 for any missing degree in the dividend, like writing x³ + 1 as x³ + 0x² + 0x + 1 — skipping a missing degree misaligns every subsequent subtraction step, the same way skipping a place value would in numeric long division.

Common Mistakes

Forgetting to distribute the quotient term across the entire divisor before subtracting.
Each step multiplies the new quotient term by the entire divisor, not just its leading term — subtract that full product from the current remainder, the same as multiplying a full divisor in numeric long division.
Making a sign error while subtracting, especially forgetting to distribute a negative sign across every term of the product being subtracted.
Rewrite a subtraction step as adding the opposite of every term being subtracted — this avoids losing track of a sign in the middle of a multi-term subtraction.

Key Takeaways

Polynomial long division mirrors numeric long division exactly, with degree playing the role of place value.
Each step divides leading terms, multiplies back through the full divisor, and subtracts.
A polynomial division can end with a nonzero remainder, expressed as dividend = (divisor)(quotient) + remainder.

Summary

Polynomial long division works, degree by degree, exactly like the numeric version you already know. The next lesson introduces a faster shortcut for the common case of dividing by a simple linear factor.

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