The polynomial division calculator above divides one polynomial by another and prints the entire long-division layout — every subtraction line, every bring-down, and the remainder at each stage. Most online dividers hand you a quotient and a remainder and leave the working invisible, which is exactly the part a student is being marked on.
Arb Digital built this as part of a free algebra set aimed at the step-by-step questions, not just the answers. Polynomial division is a procedure, and a procedure is learned by watching it run on your own numbers.
What This Polynomial Division Calculator Does
Enter the coefficients of the dividend and the divisor, highest power first. The calculator performs long division and returns the quotient polynomial, the remainder polynomial, the degree of each, and whether the division comes out exactly. Below that, it renders the traditional layout with the subtraction rows aligned by power, so you can compare it directly against your handwritten working.
Coefficients are entered as a comma-separated list. That is deliberate: it forces you to write a zero for every missing power, which is the single biggest source of error when dividing by hand.
How to Use It
- Write the dividend in descending order and list its coefficients. For 2x3 − 3x2 − 4, the list is 2, −3, 0, −4.
- Do the same for the divisor. x − 2 becomes 1, −2; x2 + 1 becomes 1, 0, 1.
- Include a zero for every missing power. A gap in the list shifts every term and produces a completely wrong answer.
- Press Divide to get the quotient, the remainder and the full layout.
- Check the result by confirming that divisor × quotient + remainder returns the dividend.
The Method: How Polynomial Long Division Works
The procedure mirrors numerical long division exactly. At each step you look only at the leading terms.
Divide the leading term of what remains of the dividend by the leading term of the divisor. That gives the next term of the quotient. Multiply the entire divisor by that quotient term. Subtract the product from the current remainder — and this is where signs go wrong most often, because subtracting a negative adds. Bring down the next term and repeat.
You stop when the degree of what is left is strictly less than the degree of the divisor. Whatever remains at that point is the remainder, and it can be a constant, a linear expression, or anything of lower degree than the divisor. The result is always expressible as dividend = divisor × quotient + remainder, which is the polynomial version of the division algorithm described in the Wolfram MathWorld entry on long division.
A Fully Worked Example
Divide 2x3 − 3x2 + 0x − 4 by x − 2.
Step 1. 2x3 ÷ x = 2x2, the first quotient term. Multiply: 2x2(x − 2) = 2x3 − 4x2. Subtract: (2x3 − 3x2) − (2x3 − 4x2) = x2. Note the sign: subtracting −4x2 adds 4x2.
Step 2. Bring down the 0x. Now divide x2 ÷ x = x. Multiply: x(x − 2) = x2 − 2x. Subtract: (x2 + 0x) − (x2 − 2x) = 2x.
Step 3. Bring down the −4. Divide 2x ÷ x = 2. Multiply: 2(x − 2) = 2x − 4. Subtract: (2x − 4) − (2x − 4) = 0.
The quotient is 2x2 + x + 2 with remainder 0. Because the remainder is zero, x − 2 is a factor, and 2x3 − 3x2 − 4 = (x − 2)(2x2 + x + 2). Multiplying that back out confirms it: 2x3 + x2 + 2x − 4x2 − 2x − 4 = 2x3 − 3x2 − 4.
Synthetic Division: The Shortcut and Its Limits
When the divisor is linear and monic — of the form x − r — synthetic division does the same job with far less writing. Write r to one side, list the dividend coefficients, bring the first one down, then repeatedly multiply by r and add to the next coefficient. The final number is the remainder and the rest are the quotient coefficients.
For the example above, r = 2 and the coefficients are 2, −3, 0, −4. Bring down 2. Then 2×2 = 4, and −3 + 4 = 1. Then 1×2 = 2, and 0 + 2 = 2. Then 2×2 = 4, and −4 + 4 = 0. Reading off: quotient 2, 1, 2 and remainder 0 — identical to the long division, in a third of the space.
The limits matter. Synthetic division only works for a linear divisor. If the leading coefficient is not 1, as in 2x − 3, you must first rewrite it as 2(x − 1.5), divide by x − 1.5, and then divide the resulting quotient by 2. For any divisor of degree two or more, synthetic division does not apply at all and long division is the only route.
The Remainder and Factor Theorems
Dividing a polynomial P(x) by x − r always leaves a constant remainder, and that constant is exactly P(r). This is the remainder theorem, and it turns a division problem into a single substitution. To find the remainder when x4 − 3x + 1 is divided by x − 2, you do not divide at all — you evaluate: 16 − 6 + 1 = 11.
The factor theorem is the special case where that remainder is zero. If P(r) = 0 then x − r divides P(x) exactly, and r is a root. This is why division and root-finding are the same activity viewed from two directions: find a root and you can divide out a factor, dropping the degree by one and making the rest of the problem easier. A cubic with one known root reduces to a quadratic you can finish with the quadratic equation solver.
Reading the Remainder as a Fraction
A non-zero remainder is usually written as a fraction on the end of the quotient. If dividing gives quotient Q(x) and remainder R(x), the full answer is Q(x) + R(x)/D(x), where D is the divisor. This form is what calculus needs for partial fractions and for finding oblique asymptotes.
The asymptote case is a good illustration of why division earns its place. A rational function whose numerator degree is exactly one more than its denominator degree has a slant asymptote, and that asymptote is simply the quotient of the division, with the remainder fraction shrinking to zero as x grows large. Divide, discard the remainder term, and the line you are left with is the asymptote — no limits required. The Wolfram MathWorld entry on synthetic division sets out the shortcut form of the same algorithm and the conditions under which it is valid, which is worth reading before relying on it in an exam.
Why Missing Powers Break Everything
The most frequent error in polynomial long division has nothing to do with the algorithm. It is writing x3 − 8 as coefficients 1, −8 instead of 1, 0, 0, −8. Skipping the zero coefficients shifts every subsequent column left, and the layout silently produces a plausible-looking but entirely wrong quotient.
The same applies to the divisor. x2 + 1 has coefficients 1, 0, 1 — the missing x term needs its zero just as much. This calculator will not guess at gaps for you, precisely because guessing hides the mistake rather than fixing the habit.
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Browse Free Tools Talk To UsCommon Mistakes to Avoid
- Omitting zero coefficients for missing powers. Every gap in the descending sequence needs a zero, in both the dividend and the divisor.
- Adding instead of subtracting. Each step subtracts the product from the running remainder, so a negative term in the product becomes a positive contribution.
- Stopping too early or too late. You continue until the remainder's degree is strictly below the divisor's degree — not until it is a constant.
- Using synthetic division with a non-linear divisor. It only works for x − r, and applying it to a quadratic divisor produces nonsense.
- Forgetting the sign of r in synthetic division. Dividing by x + 3 means r = −3, not 3.
Related Free Tools From Arb Digital
Once a factor is divided out, finish a cubic with the quadratic equation solver, and use the fraction calculator to tidy the remainder fraction. The exponent calculator helps with the power arithmetic, the LCM and GCF calculator handles common factors before you start, and the scientific calculator covers the substitution when you apply the remainder theorem. The full free online tools hub lists everything else.
Frequently Asked Questions
Write a zero for every missing power. The polynomial x cubed minus eight has coefficients 1, 0, 0, minus 8. Leaving the zeros out shifts every column and gives the wrong quotient.
They give the same answer, but synthetic division is a compressed shortcut that only works when the divisor is linear with a leading coefficient of one. Long division works for a divisor of any degree.
The remainder when a polynomial is divided by x minus r equals the value of the polynomial at r. So you can find a linear divisor's remainder by substitution instead of dividing.
When the remainder is zero. By the factor theorem that happens precisely when the divisor's root is also a root of the dividend, so the divisor is a genuine factor.
Yes. The remainder only has to be of lower degree than the divisor. Dividing by a quadratic can leave a linear remainder such as three x plus one.
Factor the leading coefficient out first, dividing by x minus 1.5, then divide the resulting quotient by 2. The remainder is unaffected by that rescaling.
Reducing the degree of an equation once a root is known, finding slant asymptotes of rational functions, and setting up partial fractions before integrating.
This calculator is provided for study and reference. Check any result by multiplying the divisor by the quotient and adding the remainder — it must reproduce the original dividend.
