Polynomial Long Division Calculator
Here are two lines from a student's paper. One of them is worth three marks and the other is worth none:
subtract (x − 7) → −x − 7 ✗
subtract (x − 7) → −x + 7 ✓
The first sign got changed and the second one got copied. That single habit costs more marks in polynomial division than everything else put together, and almost no calculator shows you the line where it happens — they print the row being subtracted and move on. The polynomial long division calculator above prints it twice: once as you write it, and once with every sign flipped, so you can see exactly what changes.
The Three Moves, Repeated
Long division of polynomials is the same loop over and over. Divide, multiply, subtract.
Divide the leading term of what is left by the leading term of the divisor. That gives the next term of the quotient — x³ ÷ x = x².
Multiply the whole divisor by that term. Not the leading term of the divisor. The whole thing.
Subtract the result. Then bring the next term down and go again.
You keep looping until what is left has a lower degree than the divisor. That is the stopping rule, and it is the only one — not "when it looks finished".
Why the Signs Flip
Subtracting a row means subtracting every term in it, because the minus sign belongs to the whole bracket and not just its first term.
−(x − 7) = −x + 7
Written out that way it looks obvious. In the middle of a division, with the rows stacked and the columns lining up, it stops being obvious — which is why the calculator shows the flipped row in red on every single step rather than leaving you to do it in your head.
If your remainder comes out with the wrong sign and you cannot find the error, this is where it is. Go back and check that every term in every subtracted row changed, not just the leading one.
Missing Terms Need a Zero
x³ − 1 has no x² term and no x term. Written out for division it is:
x³ + 0x² + 0x − 1
Four slots, not two. If you skip the gaps, the columns shift and every subtraction after that point lands under the wrong power. The arithmetic still works, which is what makes it dangerous — you get a clean-looking answer to a different question. The calculator fills the zeros in and shows them.
Dividing by x² + x + 1 and Other Higher Divisors
This is where long division earns its place. Synthetic division is quicker and less error-prone, but it only works when the divisor has degree 1. Divide by x² + x + 1 or x³ − 2 and there is no shortcut — long division is the method.
Nothing about the loop changes. You still divide leading term by leading term, still multiply the whole divisor, still subtract the whole row. The only difference is that each subtracted row is longer, so there are more signs to get wrong.
The stopping rule shifts too. Dividing by a quadratic, you stop when the leftover is linear or constant, not when it reaches a constant. x⁵ + 2x³ − x + 4 divided by x³ − 2x + 1 gives a quotient of x² + 4 and a remainder of −x² + 7x — degree 2, which is lower than the divisor's degree 3, so it stops there.
Writing the Answer
Papers ask for it two ways and it is worth being fluent in both.
| Quotient and remainder | x² − 5x + 3, remainder 4 |
| As one expression | x² − 5x + 3 + 4⁄(x − 7) |
| As an identity | x³ − 12x² + 38x − 17 = (x − 7)(x² − 5x + 3) + 4 |
The third form is the check. Multiply the divisor by the quotient, add the remainder, and you must get back exactly what you started with. The calculator runs it every time before showing an answer — if it ever failed, nothing would appear on screen at all.
How to Use This Polynomial Long Division Calculator
Put the polynomial in the first box and the divisor in the second, then press =. Terms can be in any order, brackets are expanded first, and fractions and decimals both work. Tap either box to aim the keypad at it; the key icon beside a box switches that one to your own keyboard.
The band gives the quotient and remainder, and the same answer written as a single expression underneath. Below that sits the working: for every step, the next term of the quotient, the row you multiply out, that row with its signs flipped in red, and what is left in green. The final step multiplies everything back to prove it.
Worked Examples
| Division | Quotient | Remainder | Watch for |
|---|---|---|---|
| x³ − 12x² + 38x − 17 ÷ x − 7 | x² − 5x + 3 | 4 | the plain case |
| x³ − 1 ÷ x² + x + 1 | x − 1 | 0 | degree-2 divisor, two missing terms |
| 2x⁴ + 3x² − 1 ÷ x² − 1 | 2x² + 5 | 4 | no x³ or x term to bring down |
| x⁵ + 2x³ − x + 4 ÷ x³ − 2x + 1 | x² + 4 | −x² + 7x | remainder is a quadratic, and that is correct |
| 6x³ + 5x² − 7 ÷ 2x − 1 | 3x² + 4x + 2 | −5 | leading coefficient is not 1 |
| x³ − 6x² + 11x − 6 ÷ x − 3 | x² − 3x + 2 | 0 | no remainder, so x − 3 is a factor |
That last row is the one worth remembering. A remainder of zero means the divisor divides exactly, which makes it a factor — and that is how division earns its keep in factoring and in finding zeros.
Frequently Asked Questions
How do you do long division with polynomials?
Why do the signs change when you subtract in long division?
What do you do about missing terms when dividing polynomials?
When do you stop dividing polynomials?
Can you divide by a polynomial of degree 2 or more?
How do you write the answer to a polynomial division?
Is this polynomial long division calculator free?
Long division of polynomials is not hard, it is just unforgiving. Put a zero in every gap, multiply the whole divisor rather than its first term, flip every sign when you subtract, and stop when the leftover drops below the divisor's degree. Get those four right and the method never fails you. Use the calculator above to watch each one happen until they become automatic.