Polynomial Calculator (Division and Synthetic Division)

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polynomial long division calculator

Polynomial long division calculator with steps

Pick the A ÷ B operation on the Operations tab, type both polynomials and the division appears as long division, row by row. Missing terms go in with a zero coefficient, and the remainder is checked with the identity A = B·Q + R.

How long division works, row by row

Euclidean division of A by B, with B nonzero, produces a quotient Q and a remainder R such that A = B·Q + R, where R is zero or has a smaller degree than B. Long division builds Q one term at a time. You divide the leading term of the running remainder by the leading term of B, and the result is the next term of Q. You multiply that term by the whole of B, subtract the product from the running remainder (which is why the tool writes the product with its signs flipped, underlined) and repeat with what is left.
A step fails in only one way: forgetting a missing term. In 2x³ − 3x + 1 there is no x² term, and the columns only line up if it goes in as 0x². The tool does this on its own and shows the zero in gray on the row. The number of steps is deg(A) − deg(B) + 1 when the degree of A is at least that of B, and the remainder never has a degree equal to or greater than the divisor. If the degree of A is smaller, the quotient is 0 and the remainder is A itself, as in (x² + 1) ÷ (x³ + 2).
When the divisor has a leading coefficient other than 1, fractions show up in the quotient, and the tool keeps them exact: (6x⁴ − 5x³ + 4x − 1) ÷ (2x − 1) gives quotient 3x³ − x² − (1/2)x + 7/4 and remainder 3/4, with no rounding. If the divisor has degree 1, the same division can be done with synthetic division, and the tool shows the matching table just below the long division, for comparison.

Worked divisions

Nonzero remainder

Input: (2x^3 - 3x + 1) / (x^2 + x - 2)
Expected output: Q = 2x − 2; R = 3x − 3

Two steps: 2x and −2. The x² term of the dividend goes in as 0x².

Exact division

Input: (x^3 - 6x^2 + 11x - 6) / (x - 1)
Expected output: Q = x² − 5x + 6; R = 0

Zero remainder: x − 1 is a factor and 1 is a root of the dividend.

Divisor with leading coefficient 2

Input: (6x^4 - 5x^3 + 4x - 1) / (2x - 1)
Expected output: Q = 3x³ − x² − (1/2)x + 7/4; R = 3/4

Exact fractions in the quotient, no decimals.

Degree 2 divisor, exact division

Input: (x^4 - 1) / (x^2 + 1)
Expected output: Q = x² − 1; R = 0

It amounts to x⁴ − 1 = (x² + 1)(x² − 1).

Dividend of lower degree

Input: (x^2 + 1) / (x^3 + 2)
Expected output: Q = 0; R = x² + 1

No steps: the quotient is zero and the remainder is the dividend.

Degree 1 remainder with a degree 2 divisor

Input: (4x^3 + 2x^2 - 5) / (2x^2 - 1)
Expected output: Q = 2x + 1; R = 2x − 4

The remainder 2x − 4 has degree 1, below 2, and the division stops.

Full tool FAQ

How do you divide polynomials with long division?

Divide the leading term of the dividend by the leading term of the divisor, multiply the result by the whole divisor, subtract it from the dividend and repeat with the running remainder until the remainder has a lower degree than the divisor. Example: (2x³ − 3x + 1) ÷ (x² + x − 2). First 2x³ ÷ x² = 2x; 2x·(x² + x − 2) = 2x³ + 2x² − 4x; what is left is −2x² + x + 1. Then −2x² ÷ x² = −2; −2·(x² + x − 2) = −2x² − 2x + 4; what is left is 3x − 3. Quotient 2x − 2, remainder 3x − 3. Missing terms, like the x² in the dividend, go in with a zero coefficient.

How does synthetic division work?

What is the remainder theorem and how does it help?

How do you find the roots of a polynomial of degree 3 or higher?

How do you divide when the divisor is ax + b with a different from 1?

What are Vieta's formulas?

What does the tool do with complex roots?

Can I use fractional or decimal coefficients?

How do you find the GCD of two polynomials?

What are the limits and what is saved?

Frequently asked questions

How do you know when to stop dividing polynomials?

Stop when the remainder is zero or has a lower degree than the divisor. In (4x³ + 2x² − 5) ÷ (2x² − 1), after the second step what is left is 2x − 4, of degree 1, and you can no longer divide by a term in x².

What do you do about missing terms in the dividend?

Fill in a zero coefficient, in descending order of degree. That keeps the columns aligned: x³ − 1 becomes x³ + 0x² + 0x − 1. The tool does this automatically and shows the zeros in gray.

How do you check the result of a division?

Multiply the divisor by the quotient and add the remainder: the result must be the dividend. In (x³ − 6x² + 11x − 6) ÷ (x − 1), (x − 1)(x² − 5x + 6) + 0 = x³ − 6x² + 11x − 6. The Check section does exactly this.

Next steps

How long division works, row by row

Euclidean division of A by B, with B nonzero, produces a quotient Q and a remainder R such that A = B·Q + R, where R is zero or has a smaller degree than B. Long division builds Q one term at a time. You divide the leading term of the running remainder by the leading term of B, and the result is the next term of Q. You multiply that term by the whole of B, subtract the product from the running remainder (which is why the tool writes the product with its signs flipped, underlined) and repeat with what is left.

A step fails in only one way: forgetting a missing term. In 2x³ − 3x + 1 there is no x² term, and the columns only line up if it goes in as 0x². The tool does this on its own and shows the zero in gray on the row. The number of steps is deg(A) − deg(B) + 1 when the degree of A is at least that of B, and the remainder never has a degree equal to or greater than the divisor. If the degree of A is smaller, the quotient is 0 and the remainder is A itself, as in (x² + 1) ÷ (x³ + 2).

When the divisor has a leading coefficient other than 1, fractions show up in the quotient, and the tool keeps them exact: (6x⁴ − 5x³ + 4x − 1) ÷ (2x − 1) gives quotient 3x³ − x² − (1/2)x + 7/4 and remainder 3/4, with no rounding. If the divisor has degree 1, the same division can be done with synthetic division, and the tool shows the matching table just below the long division, for comparison.

Worked divisions

Nonzero remainder

Input: (2x^3 - 3x + 1) / (x^2 + x - 2)
Expected output: Q = 2x − 2; R = 3x − 3

Two steps: 2x and −2. The x² term of the dividend goes in as 0x².

Exact division

Input: (x^3 - 6x^2 + 11x - 6) / (x - 1)
Expected output: Q = x² − 5x + 6; R = 0

Zero remainder: x − 1 is a factor and 1 is a root of the dividend.

Divisor with leading coefficient 2

Input: (6x^4 - 5x^3 + 4x - 1) / (2x - 1)
Expected output: Q = 3x³ − x² − (1/2)x + 7/4; R = 3/4

Exact fractions in the quotient, no decimals.

Degree 2 divisor, exact division

Input: (x^4 - 1) / (x^2 + 1)
Expected output: Q = x² − 1; R = 0

It amounts to x⁴ − 1 = (x² + 1)(x² − 1).

Dividend of lower degree

Input: (x^2 + 1) / (x^3 + 2)
Expected output: Q = 0; R = x² + 1

No steps: the quotient is zero and the remainder is the dividend.

Degree 1 remainder with a degree 2 divisor

Input: (4x^3 + 2x^2 - 5) / (2x^2 - 1)
Expected output: Q = 2x + 1; R = 2x − 4

The remainder 2x − 4 has degree 1, below 2, and the division stops.

Full tool FAQ

How do you divide polynomials with long division?

How does synthetic division work?

What is the remainder theorem and how does it help?

How do you find the roots of a polynomial of degree 3 or higher?

How do you divide when the divisor is ax + b with a different from 1?

What are Vieta's formulas?

What does the tool do with complex roots?

Can I use fractional or decimal coefficients?

How do you find the GCD of two polynomials?

What are the limits and what is saved?

Frequently asked questions

How do you know when to stop dividing polynomials?

What do you do about missing terms in the dividend?

Fill in a zero coefficient, in descending order of degree. That keeps the columns aligned: x³ − 1 becomes x³ + 0x² + 0x − 1. The tool does this automatically and shows the zeros in gray.