About Polynomial Long Division Calculator
This polynomial long division calculator divides one polynomial by another in the same layout as number division, and gives the quotient, the remainder and the steps. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.
The page opens on (2x³ + 3x² - 5x + 6) ÷ (x + 3). Divide the leading terms, 2x³ ÷ x = 2x², multiply 2x² by the divisor, subtract, bring down and repeat. The quotient is 2x^2 - 3x + 4 with remainder -6, which means (2x³ + 3x² - 5x + 6) = (x + 3)(2x² - 3x + 4) - 6.
If the remainder is zero the divisor is a factor, which is how long division is used to factor cubics once one root is known. By the remainder theorem, the remainder on dividing by x - a equals the polynomial's value at a.
How to use Polynomial Long Division Calculator
- Choose With remainder, With decimals or Polynomials.
- Type the dividend (the number being divided) and the divisor (the number you divide by).
- In decimal mode, choose how many decimal places you want at most.
- Read the quotient and remainder at the top, then follow the drawn layout below.
- Open the steps to see each digit: divide, multiply, subtract, bring down. The check line multiplies back to confirm.
Features
- The classic bracket layout, drawn with real columns and lines.
- Quotient and remainder for whole numbers, with the mixed-number form.
- Decimal expansion that stops when it ends or repeats, with the repeating block marked.
- Decimal dividends and divisors, with the decimal point shifted for you.
- Polynomial long division with exact fractions and the quotient plus remainder form.
- Step-by-step list and a multiplication check.
- Negative numbers and rounded results for long periods.
Polynomial division
The same relationship as for numbers holds, with the remainder of lower degree than the divisor.
P(x) = D(x) · Q(x) + R(x)
deg R < deg DWorked examples
(2x³ + 3x² - 5x + 6) ÷ (x + 3) gives 2x^2 - 3x + 4, remainder -6
The remainder equals P(-3) = -6.
(x³ - 6x² + 11x - 6) ÷ (x - 1) gives x^2 - 5x + 6, remainder 0
Zero remainder, so x - 1 is a factor.
Privacy: how it works
- Everything is calculated by code running in your browser tab. The numbers you enter are not sent to Ulkaa or to any other server, and there is nothing to sign in to.
Tips
- If the remainder is not zero when the digits run out, either write it as a remainder or put a decimal point and keep adding zeros.
- A decimal ends only if the divisor, in lowest terms, has no prime factors other than 2 and 5.
- To check any answer, multiply the quotient by the divisor and add the remainder. You should get the dividend.
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Numbers are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Numbers are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
How do I divide polynomials?
Divide the leading term of the dividend by the leading term of the divisor, multiply the result by the whole divisor, subtract, and repeat with what is left until its degree is below the divisor's.
What does a zero remainder mean?
The divisor divides the polynomial exactly, so it is a factor of it.
More Long Division Calculator pages
- Long Division CalculatorEnter a dividend and a divisor to see the long division drawn out, with the quotient, the remainder and every step.
- Long Division With RemaindersEnter two whole numbers to get the quotient and remainder, with the layout drawn out.
- Long Division With DecimalsDivide decimals or carry the division on past the point, with every step drawn.
- Repeating Decimal CalculatorDivide two numbers and see the repeating block marked, such as 1 ÷ 7 = 0.(142857).