Polynomial Long Division Calculator

What Is Polynomial Long Division?

Polynomial long division is the polynomial version of the long division you learned for numbers. Instead of dividing 742 by 7, you divide one polynomial by another and obtain a quotient and a remainder. The result is always written as: Dividend = Divisor · Quotient + Remainder. When the division is finished, the remainder has a lower degree than the divisor.

This matters because many algebra tasks require rewriting a complicated rational expression into something easier to interpret. Long division also helps when graphing rational functions, simplifying expressions before integration in calculus, and finding factors when combined with remainder ideas like the Remainder Theorem.

How the Long Division Process Works

The method repeats the same cycle: compare leading terms, divide to get the next quotient term, multiply the divisor by that term, then subtract to produce a new remainder. Each loop reduces the degree of the remainder until it is smaller than the divisor degree.

A helpful mental model is “peeling off” the highest power of the dividend using the divisor. Every subtraction removes the highest-degree term you can eliminate, which is why aligning like powers is so important.

Quotient and Remainder: What Do They Mean?

If you divide P(x) by D(x), you get Q(x) and R(x) such that: P(x) = D(x)·Q(x) + R(x). If R(x) = 0, the divisor divides evenly and D(x) is a factor of P(x). If R(x) is not zero, you can still express the division as: P(x)/D(x) = Q(x) + R(x)/D(x). This form is often used to simplify rational expressions.

Why Missing Terms Are Not a Problem

Many polynomials skip powers, like x⁴ + 1. Long division still works because the missing powers are treated as zero coefficients. In written work, you often insert placeholders (0x³, 0x², 0x) to keep terms aligned. This calculator does that automatically so you can enter the polynomial naturally.

When the Divisor Has Higher Degree Than the Dividend

If the divisor degree is greater than the dividend degree, you cannot remove the leading term of the dividend at all. In that case, the quotient is 0 and the remainder is the dividend. This is still a valid division result and is consistent with the identity Dividend = Divisor·0 + Dividend.

Synthetic Division Connection (Linear Divisors)

Synthetic division is a shortcut when your divisor is linear, especially in the form (x − r). In that case, the remainder equals f(r), and the quotient coefficients appear directly in the synthetic table. This tool can compute the equivalent r value for a general linear divisor (ax + b) by using r = −b/a and show a quick validation.

Common Mistakes and How to Avoid Them

• Not ordering terms: Put terms in descending powers before dividing.
• Sign errors: Subtraction changes signs—double-check each subtraction step.
• Forgetting missing powers: Treat missing terms as 0 to keep alignment correct.
• Stopping too early: Stop only when remainder degree is less than divisor degree.

Examples You Can Try

• (2x^3 + 3x^2 − x + 5) ÷ (x − 2)
• (x^4 − 1) ÷ (x − 1)
• (x^3 + 1) ÷ (x + 1)
• (3x^3 − 2x + 7) ÷ (x^2 + 1)

Limitations and Safe Use Notes

This calculator focuses on long division of expanded polynomials in one variable. If you enter parentheses, expand first for the most predictable results. Exact mode returns simplified fractions when coefficients are rational; decimal mode rounds values to the selected precision for readability.