Polynomial Division Calculator
Use this polynomial division calculator to divide one polynomial by another and see the quotient, remainder, and long division steps.
Quick Result Explanation
The calculator rewrites the division in the form dividend = divisor times quotient plus remainder. If the remainder is 0, the divisor divides the dividend exactly. If a remainder remains, the quotient is only part of the result.
Polynomial Division Formula and Method
Polynomial long division follows the same idea as number long division: divide the leading term, multiply back, subtract, and repeat until the remaining polynomial has a lower degree than the divisor.
- P(x) is the dividend polynomial.
- D(x) is the divisor polynomial.
- Q(x) is the quotient polynomial.
- R(x) is the remainder polynomial, and its degree must be lower than the degree of the divisor.
How to Use the Polynomial Division Calculator
- Enter the dividend polynomial in the first box.
- Enter the divisor polynomial in the second box.
- Use x as the variable, such as x^3 + 2x – 5.
- Choose how many decimal places to show if coefficients are not whole numbers.
- Select Calculate Division to get the quotient, remainder, and steps.
- Use Copy Result if you want to save the visible answer.
Worked Example
Divide x^3 – 2x^2 + 4x – 8 by x – 2.
| Step | Action |
|---|---|
| 1 | Divide the leading terms: x^3 divided by x = x^2. |
| 2 | Multiply x^2 by x – 2, then subtract it from the dividend. |
| 3 | Repeat with the new leading term until the remainder has lower degree than x – 2. |
| Result | The quotient is x^2 + 4 and the remainder is 0. |
Common Use Cases
- Checking algebra homework that involves polynomial long division.
- Finding a quotient and remainder before factoring a polynomial.
- Testing whether a linear expression is a factor of a polynomial.
- Preparing for synthetic division by first seeing the long division structure.
- Simplifying rational polynomial expressions in algebra and precalculus.
Common Mistakes
- Forgetting missing terms, such as entering x^3 + 1 when you meant x^3 + 0x^2 + 0x + 1. The calculator can handle missing terms, but the steps may look shorter.
- Using a different variable such as y. This calculator uses x only.
- Entering negative exponents. Polynomial division requires nonnegative whole number exponents.
- Dividing by 0 or by an empty divisor. A valid divisor polynomial is required.
- Mixing multiplication symbols in unusual ways. Use 3x^2 instead of 3*x^2 when possible.
Limitations
This calculator handles one-variable polynomials in x with real-number coefficients and nonnegative integer exponents. It does not solve equations, factor polynomials completely, or perform multivariable polynomial division. Decimal rounding only affects the displayed result, not the internal long division calculation.
Related Calculators
These tools can help with nearby algebra and arithmetic tasks:
Polynomial Division FAQ
What is polynomial division?
Polynomial division is the process of dividing one polynomial by another to find a quotient and a remainder.
What does the remainder mean?
The remainder is what is left after division stops. Its degree must be lower than the degree of the divisor.
Can this calculator show steps?
Yes. It shows each leading-term division, the polynomial being subtracted, and the new remainder.
Can I use decimals in coefficients?
Yes. You can enter coefficients such as 0.5x^2 or 2.75x, then choose how many decimal places to display.
Is synthetic division the same as polynomial long division?
No. Synthetic division is a shortcut for certain divisors, usually linear divisors. Long division works for a wider range of polynomial divisors.
