polynomial long division calculator

What this polynomial long division calculator does

This calculator divides one polynomial by another using the same process you learn in algebra class: divide leading terms, multiply, subtract, and repeat. You get the quotient, the remainder, and a readable set of steps so you can verify your work.

It is useful for homework checking, exam prep, and quick factorization testing. If the remainder is zero, your divisor is an exact factor of the dividend.

How to use the calculator

1) Enter the dividend and divisor

Type each polynomial in standard algebra format. The variable is x, and powers should use ^ (for example, x^3).

2) Click Calculate Division

The calculator returns:

• The quotient polynomial
• The remainder polynomial
• A verification identity: Dividend = Divisor × Quotient + Remainder
• Step-by-step subtraction rounds from the long division process

Accepted input format

Use clear algebraic terms separated by + or -. Examples:

• x^3 - 4x + 7
• 2x^4 + x^2 - 9
• -3x^2 + x - 1
• 5 (a constant polynomial)

Decimals are supported (such as 0.5x^2 - 1.25x + 3). Negative exponents and parentheses are not supported in this version.

Quick refresher: polynomial long division

The core loop

• Divide the highest-degree term in the remainder by the highest-degree term in the divisor.
• Write that term in the quotient.
• Multiply the whole divisor by that term.
• Subtract from the current remainder.
• Repeat until the remainder degree is lower than the divisor degree.

The final answer always has the form: Quotient + (Remainder / Divisor).

Why the remainder matters

The remainder gives important structural information:

• If remainder is 0, the divisor is a factor.
• For division by x - a, the remainder equals f(a) (Remainder Theorem).
• Nonzero remainders help identify near-factors and simplify rational expressions correctly.

Common mistakes this tool helps you catch

• Forgetting missing terms (like skipping 0x^2 mentally)
• Sign errors during subtraction
• Incorrect leading-term division
• Stopping before remainder degree is small enough

Practice ideas

Start by dividing cubics by linear terms, then move to quartics by quadratics. Try problems where the remainder is zero and nonzero so you can recognize both patterns quickly.