Polynomial Long Division Calculator

What this polynomial long division calculator does

This polynomial long division calculator divides one polynomial (the dividend) by another (the divisor) and returns the quotient and remainder in standard algebraic form. It follows the same step pattern as numerical long division: divide, multiply, subtract, and bring down the next term.

The tool is designed for typical Algebra 2, Pre‑Calculus, and introductory college algebra courses where you learn how to divide polynomials, simplify rational expressions, and prepare for standardized tests. By automating the arithmetic, you can check homework, confirm exam practice problems, and explore more complex examples without getting lost in the algebra.

How to use the polynomial long division calculator

1. Enter the dividend polynomial. Type the polynomial you want to divide in the first input, for example 2x^3 + 3x^2 - 5x + 6.
2. Enter the divisor polynomial. Type the polynomial you are dividing by in the second input, for example x - 2 or x^2 + x - 1.
3. Use standard notation. Use x as the variable, ^ for powers (like x^3), and + or - between terms. Spaces are optional.
4. Run the division. Click the Divide button to compute the quotient and remainder. The result appears in a results panel below the form.
5. Copy your answer. Use the Copy result button (if available) to paste the simplified expression into your notes or assignments.

The polynomial long division formula

Polynomial long division is based on the idea that, for a nonzero divisor D(x), any polynomial P(x) can be written in the form

where:

• P(x) is the dividend polynomial.
• D(x) is the divisor polynomial.
• Q(x) is the quotient polynomial.
• R(x) is the remainder polynomial.

The degree of the remainder is always less than the degree of the divisor. If the remainder is the zero polynomial, then the divisor divides the dividend exactly.

In terms of a rational expression, you can also write

This is exactly what the calculator computes: the quotient Q(x) and the remainder R(x) that satisfy this relationship.

Step-by-step algorithm (what the calculator does)

The calculator implements the standard long division algorithm. Conceptually, it follows these steps:

1. Arrange both polynomials in descending powers of x, inserting missing powers with coefficient 0 (for example, write x^3 + 6 as x^3 + 0x^2 + 0x + 6).
2. Divide the leading term of the dividend by the leading term of the divisor to get the next term of the quotient.
3. Multiply the entire divisor by this quotient term.
4. Subtract the result from the current dividend to form a new remainder polynomial.
5. Repeat from step 2 using the new remainder as the dividend, stopping when the degree of the remainder is less than the degree of the divisor.

Worked example: dividing by a linear polynomial

Suppose we want to divide 2x^3 + 3x^2 - 5x + 6 by x - 2.

1. Compare leading terms. Divide 2x^3 by x to get 2x^2. This is the first term of the quotient.
2. Multiply and subtract. Multiply the divisor by 2x^2 to get 2x^3 - 4x^2. Subtracting this from the original dividend gives a new polynomial: (2x^3 + 3x^2 - 5x + 6) - (2x^3 - 4x^2) = 7x^2 - 5x + 6.
3. Repeat the process. Now divide 7x^2 by x to get 7x. Multiply the divisor by 7x to get 7x^2 - 14x. Subtracting gives (7x^2 - 5x + 6) - (7x^2 - 14x) = 9x + 6.
4. One more iteration. Divide 9x by x to get 9. Multiply the divisor by 9 to get 9x - 18. Subtracting gives (9x + 6) - (9x - 18) = 24.
5. Stop condition. The remainder 24 has degree 0, which is less than the degree 1 of x - 2, so the algorithm stops.

The final answer is 2x^3 + 3x^2 - 5x + 6 = (x - 2)(2x^2 + 7x + 9) + 24. The calculator will show the quotient 2x^2 + 7x + 9 and the remainder 24.

Interpreting the calculator output

When you run the calculation, you typically see three key pieces of information:

• Quotient: the polynomial Q(x) such that P(x) = Q(x)D(x) + R(x).
• Remainder: the polynomial R(x) with degree less than that of the divisor.
• Combined expression: sometimes shown as Q(x) + R(x) / D(x) to emphasize the rational function form.

For example, if you divide x^3 + 6 by x - 1 the result may appear as Q(x) = x^2 + x + 1 and R(x) = 7. Interpreted as a rational expression, this means (x^3 + 6) / (x - 1) = x^2 + x + 1 + 7 / (x - 1).

Comparison with related methods

Polynomial long division is closely related to other algebraic techniques. The table below summarizes how it compares to common alternatives.

Assumptions and limitations of this calculator

To keep the algorithm fast and reliable in a browser, the calculator makes several assumptions about the polynomials you enter.

• Integer coefficients. Coefficients are assumed to be integers (for example, 3x^2 - 5x + 2). Some simple fractions or decimals may work, but they are not guaranteed.
• Non-negative exponents. Only terms like x^3, x^2, x, and constants are supported. Negative or fractional exponents (such as x^-1 or x^(1/2)) are not handled.
• Single variable. Use one variable (usually x). Multivariable expressions such as x^2 + y are outside the scope of this tool.
• Standard syntax. Use ^ for powers (for example, 4x^5), write multiplication implicitly as in 4x or explicitly as 4*x, and separate terms with + or -. Extra spaces are ignored.
• Nonzero divisor. The divisor polynomial must not be the zero polynomial. If the divisor simplifies to zero, no division is performed and you will see an error or no result.
• Handling missing terms. You do not have to type terms with zero coefficients; the calculator can infer them. For example, entering x^3 + 6 is treated as x^3 + 0x^2 + 0x + 6 during the computation.
• Invalid input. If an input cannot be parsed as a polynomial (for example, due to unmatched parentheses or unsupported symbols), the calculator will not return a result. Correct the expression and try again.

These assumptions match most textbook problems. If you need more general symbolic manipulation, such as expressions with multiple variables, fractional exponents, or exact rational coefficients, a full computer algebra system may be more appropriate.

Educational uses and next steps

Polynomial long division appears throughout the secondary and early tertiary curriculum. It is especially important when simplifying rational functions, analyzing asymptotic behavior, and preparing for topics like partial fraction decomposition and power series in calculus.

After you are comfortable using this calculator, you might also explore related tools such as a synthetic division calculator for quick division by linear factors, or a factoring polynomials calculator to help you simplify rational expressions before or after division.