What is polynomial long division?
Polynomial long division is a method for dividing one polynomial by another. It works similarly to ordinary long division with numbers: divide the leading terms, multiply, subtract, and repeat until the remaining polynomial has a lower degree than the divisor.
How to use this polynomial long division calculator
Enter the dividend and divisor, then select Divide polynomials. The calculator returns the quotient and remainder and shows the main division steps. It is useful for checking algebra homework, simplifying rational expressions, and verifying factoring results.
Polynomial division formula
Dividend = (Divisor × Quotient) + Remainder
The remainder must have a lower degree than the divisor.
Example: divide x³ + 2x² - 5x + 6 by x - 2
(x3 + 2x2 - 5x + 6) ÷ (x - 2)
Quotient = x2 + 4x + 3
Remainder = 12
So the result can be written as x2 + 4x + 3 + 12/(x - 2).
Step-by-step polynomial long division
- Divide the leading terms: x3 ÷ x = x2. This is the first term of the quotient.
- Multiply: x2(x - 2) = x3 - 2x2.
- Subtract: (x3 + 2x2 - 5x + 6) - (x3 - 2x2) = 4x2 - 5x + 6.
- Repeat: 4x2 ÷ x = 4x. Multiply 4x(x - 2) = 4x2 - 8x, then subtract to get 3x + 6.
- Finish: 3x ÷ x = 3. Multiply 3(x - 2) = 3x - 6, then subtract. The remainder is 12.
Dividing polynomials by a monomial
When the divisor has only one term, divide every term of the dividend by that term separately.
(2x3 - 5x2 + x) ÷ x = 2x2 - 5x + 1
(20x3y3 + 20x2y2) ÷ (-5xy) = -4x2y2 - 4xy
This calculator works with polynomials in one variable, x. For multi-variable monomial division, divide the coefficients and subtract the exponents of matching variables.
Synthetic division vs long division
Synthetic division is a shortcut for dividing a polynomial by a linear divisor of the form x - c. It uses only the coefficients and is usually faster than long division.
For example, to divide x3 + 2x2 - 5x + 6 by x - 2, use 2 in synthetic division:
Coefficients: 1, 2, -5, 6
Bring down 1. Multiply by 2 to get 2. Add to 2 to get 4. Multiply by 2 to get 8. Add to -5 to get 3. Multiply by 2 to get 6. Add to 6 to get 12.
Quotient: x2 + 4x + 3
Remainder: 12
Use polynomial long division when the divisor has more than one term or is not linear. Use synthetic division when the divisor is x - c.
More worked examples
(x2 - 16) ÷ (x + 4) = x - 4, remainder 0.
(x2 - 12x + 20) ÷ (x - 10) = x - 2, remainder 0.
(x2 - 49) ÷ (x + 7) = x - 7, remainder 0.
(x2 - x - 12) ÷ (x + 3) = x - 4, remainder 0.
Polynomial division homework examples
These are typical “find the quotient” problems, worked step by step. In each case divide every term of the top polynomial by the divisor.
- Find the quotient of 20x³y³ + 20x²y² divided by −5xy: 20x³y³ ÷ (−5xy) = −4x²y² and 20x²y² ÷ (−5xy) = −4xy. The quotient is −4x²y² − 4xy.
- Find the quotient of −8x⁵ + 32x⁴ + 24x³ + 12x² divided by −4x²: −8x⁵ ÷ (−4x²) = 2x³, 32x⁴ ÷ (−4x²) = −8x², 24x³ ÷ (−4x²) = −6x and 12x² ÷ (−4x²) = −3. The quotient is 2x³ − 8x² − 6x − 3.
- Divide the polynomials (x² − 49) ÷ (x + 7): x² − 49 = (x − 7)(x + 7), so the quotient is x − 7 with remainder 0.
When the divisor is a single term (a monomial), no long division is needed. When the divisor has two or more terms, use the long division steps above.
Common mistakes in polynomial division
- Forgetting to write missing powers with a coefficient of zero, such as
0x^2. - Adding instead of subtracting after multiplying the divisor term.
- Stopping before the remainder has a lower degree than the divisor.
- Using synthetic division with a divisor that is not linear.
- Forgetting that a remainder of zero means the divisor is a factor of the dividend.
Frequently asked questions
How do I divide polynomials using long division?
Divide the leading term of the dividend by the leading term of the divisor, multiply the divisor by that result, subtract, and repeat with the new remainder. Continue until the remainder has a lower degree than the divisor.
What is the difference between polynomial long division and synthetic division?
Long division works for any polynomial divisor. Synthetic division is a shortcut that works only when dividing by a linear divisor such as x - 2 or x + 4.
Can this calculator show steps?
Yes. After entering a dividend and divisor, the calculator shows the quotient, remainder, and the main long-division steps.
Can I divide polynomials with missing powers?
Yes. Missing powers are treated as terms with a coefficient of zero. For example, x³ + 6 can be treated as x³ + 0x² + 0x + 6.
What does a remainder of zero mean?
A remainder of zero means the divisor divides the dividend evenly. In other words, the divisor is a factor of the dividend.
How do I divide a polynomial by a monomial?
Divide each term of the polynomial by the monomial. For example, (6x³ + 9x²) ÷ 3x = 2x² + 3x.
Can I divide by a constant?
Yes. Dividing by a non-zero constant divides every coefficient by that constant. For example, (4x² + 8x) ÷ 2 = 2x² + 4x.
Does this calculator support decimals?
Yes. You can enter decimal coefficients, such as 1.5x^2 - 2.5x + 3.
How do you find the quotient of two polynomials?
If the divisor is a monomial, divide each term of the polynomial by it. For example, (−8x⁵ + 32x⁴ + 24x³ + 12x²) ÷ (−4x²) = 2x³ − 8x² − 6x − 3. If the divisor has more than one term, use polynomial long division and keep going until the remainder has a lower degree than the divisor.