Long Division Calculator with Variables

Variables calculator: enter polynomials in x (e.g. x^2+5x+6) ÷ (x+2).

Output: algebraic quotient and remainder polynomial.

The Long Division Calculator with Variables divides algebraic expressions that contain letters and exponents, such as x, x², and 2x + 3. It uses the same long division algorithm as with numbers, but each step works on terms: divide the leading terms, multiply the whole divisor by that quotient term, subtract, and bring down the next term. The calculator lines up like terms by degree, subtracts carefully, and reports the quotient expression with any remainder. Enter a dividend and divisor in x to see each divide–multiply–subtract step and the final quotient and remainder.

How to use the Long Division Calculator with Variables

To divide algebraic expressions, follow these 4 steps:

Enter the dividend expression, such as x² + 5x + 6, in the first field.
Enter the divisor expression, such as x + 2, in the second field.
Click Calculate to divide term by term.
Read the quotient expression, the remainder, and each step shown.

Write missing powers with a zero coefficient, such as x² + 0x − 4, so the calculator keeps every column of like terms aligned.

Dividing with variables

A variable expression is built from terms, each a coefficient times a power of the variable. Division of expressions mirrors number division because place value is replaced by the power of the variable: the highest power plays the role of the leftmost digit. You always work from the highest degree down, dividing the leading term of the current dividend by the leading term of the divisor to get the next quotient term. The Long Division Calculator with Variables applies the exponent quotient rule and keeps like terms in the same column so subtraction is exact.

Division shares a total into equal groups. Long division does this digit by digit.

Divisor (32) — the number you divide by. Place it to the left of the bracket.

How variables work in long division

The calculator divides expressions through five internal actions:

Orders both expressions from the highest power of the variable to the lowest.
Divides the leading term of the dividend by the leading term of the divisor using the exponent rule xᵐ ÷ xⁿ = xᵐ⁻ⁿ.
Multiplies the entire divisor by that quotient term.
Subtracts the product from the current dividend, cancelling the leading term.
Brings down the next term and repeats until the remainder's degree is below the divisor's.

The process stops when the leftover expression has a lower degree than the divisor, and that leftover is the remainder.

Orders both expressions from the highest power of the variable to the lowest.

Formula for dividing expressions

The Long Division Calculator with Variables uses the division identity in expression form: Dividend = Divisor × Quotient + Remainder, where deg(Remainder) < deg(Divisor). The key term rule is the exponent quotient law xᵐ ÷ xⁿ = xᵐ⁻ⁿ. For (x² + 5x + 6) ÷ (x + 2) = x + 3, the identity confirms (x + 2)(x + 3) = x² + 5x + 6.

Dividend

Divisor

Quotient

Remainder

100 = 7 × 14 + 2 → Identity holds

Algebra example problems

These examples divide expressions in x term by term.

Example 1 — Exact division: (x² + 5x + 6) ÷ (x + 2)

Divide x² ÷ x = x; multiply x(x + 2) = x² + 2x; subtract to leave 3x + 6.
Divide 3x ÷ x = 3; multiply 3(x + 2) = 3x + 6; subtract to leave 0.
So (x² + 5x + 6) ÷ (x + 2) = x + 3 with no remainder.

Example 2 — With a remainder: (x² + 1) ÷ (x − 1)

Write x² + 0x + 1; divide x² ÷ x = x; multiply x(x − 1) = x² − x; subtract to leave x + 1.
Divide x ÷ x = 1; multiply 1(x − 1) = x − 1; subtract to leave 2.
So (x² + 1) ÷ (x − 1) = x + 1 with remainder 2.

Example 3 — Monomial divisor: (6x³ + 4x²) ÷ 2x

Divide 6x³ ÷ 2x = 3x²; divide 4x² ÷ 2x = 2x.
Each term divides exactly, so there is no remainder.
So (6x³ + 4x²) ÷ 2x = 3x² + 2x.

Divide x² ÷ x = x; multiply x(x + 2) = x² + 2x; subtract to leave 3x + 6.
Divide 3x ÷ x = 3; multiply 3(x + 2) = 3x + 6; subtract to leave 0.
So (x² + 5x + 6) ÷ (x + 2) = x + 3 with no remainder.

Worked algebra problems

How do you divide 2x² + 7x + 3 by x + 3?

(2x² + 7x + 3) ÷ (x + 3) = 2x + 1. Divide 2x² ÷ x = 2x; multiply 2x(x + 3) = 2x² + 6x; subtract to leave x + 3. Divide x ÷ x = 1; multiply 1(x + 3) = x + 3; subtract to leave 0. Check: (x + 3)(2x + 1) = 2x² + 7x + 3.

x^2+5x+6x+2

Why write x² + 0x − 4 with a zero term?

A zero placeholder keeps like terms aligned so subtraction stays correct. For (x² − 4) ÷ (x − 2), writing x² + 0x − 4 gives a column for the missing x term. Dividing gives x + 2 exactly, because (x − 2)(x + 2) = x² − 4. Skipping the placeholder would misalign the columns and break the subtraction.

x^2-9x-3

Common algebra division mistakes

Dividing expressions produces 5 frequent errors:

Forgetting a zero placeholder for a missing power, which misaligns like terms.
Multiplying the quotient term by only part of the divisor.
Sign errors when subtracting the product from the current dividend.
Adding exponents instead of subtracting them when dividing terms.
Stopping before the remainder's degree drops below the divisor's degree.

The Long Division Calculator with Variables inserts placeholders, applies the exponent rule, and subtracts each product exactly.

Forgetting a zero placeholder for a missing power, which misaligns like terms.

Multiplying the quotient term by only part of the divisor.

Sign errors when subtracting the product from the current dividend.

Adding exponents instead of subtracting them when dividing terms.

Stopping before the remainder's degree drops below the divisor's degree.

Frequently Asked Questions

How do you do long division with variables?

Divide the leading term of the dividend by the leading term of the divisor, multiply the whole divisor by that quotient term, subtract, then bring down the next term and repeat until the leftover has a lower degree than the divisor.

What is the exponent rule when dividing terms?

Subtract the exponents: xᵐ ÷ xⁿ = xᵐ⁻ⁿ. For example, 6x³ ÷ 2x = 3x², dividing the coefficients and subtracting the powers.

Why do I add a zero term for a missing power?

A zero placeholder such as x² + 0x − 4 keeps each degree in its own column so the subtraction lines up correctly, just like place value in numeric long division.

When does dividing expressions leave a remainder?

It leaves a remainder when the divisor does not divide the dividend evenly. Division stops once the leftover expression has a lower degree than the divisor, and that leftover is the remainder.

How do I divide by a single term like 2x?

Divide each term of the dividend by the monomial separately. For (6x³ + 4x²) ÷ 2x, divide each term to get 3x² + 2x.

How do I check an algebraic division answer?

Multiply the divisor by the quotient and add the remainder; the result should equal the dividend. For (x² + 1) ÷ (x − 1) = x + 1 R2, check (x − 1)(x + 1) + 2 = x² + 1.

Is dividing with variables the same as polynomial division?

Yes — dividing expressions in one variable is polynomial long division. This page focuses on the term-by-term method and the exponent rule that make it work.

The identity for expressions

The Long Division Calculator with Variables satisfies Dividend = Divisor × Quotient + Remainder with deg(Remainder) < deg(Divisor), using the exponent rule xᵐ ÷ xⁿ = xᵐ⁻ⁿ at each step. For (2x² + 7x + 3) ÷ (x + 3) = 2x + 1, the identity confirms (x + 3)(2x + 1) = 2x² + 7x + 3.